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Every convergent is nearer to the value of the whole fraction than any preceding convergent.

1816It is not the quantity but the quality of the anatomical and bionomic characters which determines their taxonomic value, and a few fundamental characters are better indications of the affinities of given groups of birds than a great number of agreements if these can be shown to be cases of isomorphism or heterophyletic, convergent analogy.

1011(ii) Each convergent is a fraction in its lowest terms.

96When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency - the image of the original radiant point, the calculation assumes a less complicated form.

910(-) 5 + &c., in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.

911(ii) Each convergent is a fraction in its lowest terms.

87It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

88By separation of real and imaginary parts, C =M cos 27rv 2 +N sin 27rv2 1 S =M sin 27rv 2 - N cos 27rv2 where 35+357.9 N _ 7rv 3 7r 3 v 7 + 1.3 1.3.5.7 1.3.5.7.9.11 These series are convergent for all values of v, but are practically useful only when v is small .

89.; the series, however, is convergent for real values of x only when x lies between +I and -1.

89The resemblance shows various grades of completeness; and the convergent mimics may be themselves noxious, or edible and innocuous.

77The outermost hyphae may even put forth thinner hyphae, radiating into the soil like root-hairs, and the convergent tips may be closely appressed and so divided by septa as to resemble the root-apex of a higher plant (Armillaria mellea).

77The resemblance shows various grades of completeness; and the convergent mimics may be themselves noxious, or edible and innocuous.

77But, while Robertson was in some measure the initiator of a movement, Prescott came to his task when the range of information was incomparably wider and when progress in sociologic theory had thrown innumerable convergent lights upon the progress of events.

78This being so, it would be premature to disregard the convergent lines of historical evidence which tell against A.D.

78the former of these equations gives a convergent series for logep, and the latter a very convergent series by means of which the logarithm of any number may be deduced from the logarithm of the preceding number.

78From the formula for log e (p/q) we may deduce the following very convergent series for log e 2, log e 3 and log e 5, viz.: log e 2=2(7P +5Q +3R), log e 3 =2(11P+8Q +5R), log e 5=2(16P+12Q+7R), where P 1 +3?1) 3 5 s (31) &C.

79We may require to represent the infinite convergent power series ao+alx+ a2x 2 + ...

65Its n th convergent is not equal to the sum to n terms of the series.

65The Japanese attack was convergent, but there was no room for envelopment; the Russian position moreover was " all-round " and presented no flanks, and except for the enfilade fire of the Japanese and Russian gunboats in the shallow bays on either side the battle was locally at every point a frontal attack and defence.

66It is doubtful whether the convergent action of the streams has been the sole agency in the erosion of these striking cavities, or whether snow and glacier-ice have had a share in the work.

66(iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent.

66In this case the sum to n terms of the series is equal to the nth convergent of the fraction.

54the series on the right-hand being convergent for all values of x and therefore defining an analytical function of x which is uniform and regular all over the plane.

55The numerators and denominators of the successive convergents obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

55In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.

55- A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures.

56On the 25th of August the 2nd and 4th Armies from Haicheng and the 1st Army from the Yin-tsu-ling and Yu-shu-ling began the last stage of their convergent advance.

56Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series.

56The optic figure seen in convergent polarized light through a section cut parallel to the plane of symmetry of a borax crystal is symmetrical only with respect to the central point.

56On the 25th of August the 2nd and 4th Armies from Haicheng and the 1st Army from the Yin-tsu-ling and Yu-shu-ling began the last stage of their convergent advance.

56Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.

44- Lateral metacarpals as in Cervus; antlers small, with a brow-tine and an unbranched beam, supported on long bony pedicles, continued downwards as convergent ridges on the forehead; upper canines of male large and tusk-like.

44The two legs of a parabolic branch may converge to ultimate parallelism, as in the conic parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola y 2 = x 3, and the branch is said to be convergent, or divergent, accordingly; or they may tend to parallelism in opposite senses, as in the cubical parabola y = x 3 .

44Ina word, wood-100+10 &c., in (6) and 2-?-4+a &c., in (5) are convergent series, and I I I I and I are the limits to which they respectively approximate.] (7) So long as anything is in one and the same space, it is at rest.

44If then the objective tube is directed to any star, the convergent beam from the object-glass is received by the plane mirror from which it is reflected upwards along the polar axis and viewed through the hollow upper pivot.

46If then the objective tube is directed to any star, the convergent beam from the object-glass is received by the plane mirror from which it is reflected upwards along the polar axis and viewed through the hollow upper pivot.

46The general continued fraction al is evidently equal, convergent by convergent, to the continued fraction X 2 b 2 X2X3b3 x3%4b4 a1+ A2a2 + X + X

33Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.

33The difference between the continued fraction and the nth convergent is less than, and greater than a n+2 These limits qn may be replaced by the following, which, though not so close, are simpler, viz.

33If we suppose alb to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq-bp= +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please.

33Man and monkeys alone possess parallel and convergent vision of the two eyes, while a divergent, and consequently a very widely extended, vision is a prerogative of the lower mammals; squirrels, for instance, and probably also hares and rabbits, being able to see an object approaching them directly from behind without turning their heads.

33There is nearly always an arrangement to observe the preparation first in convergent light and then in parallel polarized light.

33The general continued fraction al is evidently equal, convergent by convergent, to the continued fraction X 2 b 2 X2X3b3 x3%4b4 a1+ A2a2 + X + X

33A continued fraction may always be found whose n th convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product.

36, p, ,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I +bnpn -2 1 qn = anq,i l +bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 + b2= q2.

25It is always possible to find the value of the n th convergent to a recurring continued fraction.

288When we can solve this equation we have an expression for the n th convergent to the fraction, generally in the form of the quotient of two series, each of n terms. As an example, take the fraction (known as Brouncker's fraction, after Lord Brouncker) I I 2325 2 72 2 + 2 + 2 + 2 + ...

120The tests for convergency are as follows: Let the continued fraction of the first class be reduced to the form dl+d2 +d3+d4+ then it is convergent if at least one of the series.

1303are Bernoulli's numbers (ï¿½ 46 (v.)), although the series is not convergent.

00The mean brightness varies as .51; and the integral f J02(z)zdz is not convergent.

00Convergent beam electron diffraction In many materials, the crystal structure of minority phases is not known.

00holomorphic function, forming a logarithm of, is defined by where Both series are absolutely convergent.

00Note that the funnel in Figure 4 is also a convergent nozzle.

00Establish convergent stonewort at a further site within its historic range by 2005.

00It is not the quantity but the quality of the anatomical and bionomic characters which determines their taxonomic value, and a few fundamental characters are better indications of the affinities of given groups of birds than a great number of agreements if these can be shown to be cases of isomorphism or heterophyletic, convergent analogy.

00But, while Robertson was in some measure the initiator of a movement, Prescott came to his task when the range of information was incomparably wider and when progress in sociologic theory had thrown innumerable convergent lights upon the progress of events.

00The acute bisectrix of the optic axes never deviates from the normal to the basal plane by more than a degree or two, hence a cleavage flake of mica will always show an optic figure in convergent light when placed on the stage of a polarizing microscope.

00are Bernoulli's numbers (Ã¯¿½ 46 (v.)), although the series is not convergent.

00- A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures.

00When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency - the image of the original radiant point, the calculation assumes a less complicated form.

00The mean brightness varies as .51; and the integral f J02(z)zdz is not convergent.

00By separation of real and imaginary parts, C =M cos 27rv 2 +N sin 27rv2 1 S =M sin 27rv 2 - N cos 27rv2 where 35+357.9 N _ 7rv 3 7r 3 v 7 + 1.3 1.3.5.7 1.3.5.7.9.11 These series are convergent for all values of v, but are practically useful only when v is small .

00The Japanese attack was convergent, but there was no room for envelopment; the Russian position moreover was " all-round " and presented no flanks, and except for the enfilade fire of the Japanese and Russian gunboats in the shallow bays on either side the battle was locally at every point a frontal attack and defence.

00This being so, it would be premature to disregard the convergent lines of historical evidence which tell against A.D.

00.; the series, however, is convergent for real values of x only when x lies between +I and -1.

00It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation log(I +x) =x -",, x2 +3x 3 -4x 4 + is true only when the analytical modulus of x is less than unity.

00the series on the right-hand being convergent for all values of x and therefore defining an analytical function of x which is uniform and regular all over the plane.

00(-) 5 + &c., in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.

00the former of these equations gives a convergent series for logep, and the latter a very convergent series by means of which the logarithm of any number may be deduced from the logarithm of the preceding number.

00From the formula for log e (p/q) we may deduce the following very convergent series for log e 2, log e 3 and log e 5, viz.: log e 2=2(7P +5Q +3R), log e 3 =2(11P+8Q +5R), log e 5=2(16P+12Q+7R), where P 1 +3?1) 3 5 s (31) &C.

00- Analogous or convergent evolution in Fish, Reptile and Mammal.

00Molar series strongly convergent behind (see Prairie-Marmot).

00The outermost hyphae may even put forth thinner hyphae, radiating into the soil like root-hairs, and the convergent tips may be closely appressed and so divided by septa as to resemble the root-apex of a higher plant (Armillaria mellea).

00It is doubtful whether the convergent action of the streams has been the sole agency in the erosion of these striking cavities, or whether snow and glacier-ice have had a share in the work.

00The numerators and denominators of the successive convergents obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

00Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.

00Every convergent is nearer to the value of the whole fraction than any preceding convergent.

00Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.

00The difference between the continued fraction and the nth convergent is less than, and greater than a n+2 These limits qn may be replaced by the following, which, though not so close, are simpler, viz.

00If we suppose alb to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq-bp= +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please.

00In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.

00When we can solve this equation we have an expression for the n th convergent to the fraction, generally in the form of the quotient of two series, each of n terms. As an example, take the fraction (known as Brouncker's fraction, after Lord Brouncker) I I 2325 2 72 2 + 2 + 2 + 2 + ...

00It is always possible to find the value of the n th convergent to a recurring continued fraction.

00The tests for convergency are as follows: Let the continued fraction of the first class be reduced to the form dl+d2 +d3+d4+ then it is convergent if at least one of the series.

00A continued fraction may always be found whose n th convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product.

00, p, ,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I +bnpn -2 1 qn = anq,i l +bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 + b2= q2.

00In this case the sum to n terms of the series is equal to the nth convergent of the fraction.

00We may require to represent the infinite convergent power series ao+alx+ a2x 2 + ...

00Its n th convergent is not equal to the sum to n terms of the series.

00Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series.

00The optic figure seen in convergent polarized light through a section cut parallel to the plane of symmetry of a borax crystal is symmetrical only with respect to the central point.

00- Lateral metacarpals as in Cervus; antlers small, with a brow-tine and an unbranched beam, supported on long bony pedicles, continued downwards as convergent ridges on the forehead; upper canines of male large and tusk-like.

00If a is zero, we may regard as the first convergent, and precede the series by and Ã‚°-.

00(iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent.

00The two legs of a parabolic branch may converge to ultimate parallelism, as in the conic parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola y 2 = x 3, and the branch is said to be convergent, or divergent, accordingly; or they may tend to parallelism in opposite senses, as in the cubical parabola y = x 3 .

00Man and monkeys alone possess parallel and convergent vision of the two eyes, while a divergent, and consequently a very widely extended, vision is a prerogative of the lower mammals; squirrels, for instance, and probably also hares and rabbits, being able to see an object approaching them directly from behind without turning their heads.

00Ina word, wood-100+10 &c., in (6) and 2-?-4+a &c., in (5) are convergent series, and I I I I and I are the limits to which they respectively approximate.] (7) So long as anything is in one and the same space, it is at rest.

00There is nearly always an arrangement to observe the preparation first in convergent light and then in parallel polarized light.

00Establish convergent stonewort at a further site within its historic range by 2005.

00Standard intelligence tests measure convergent thinking, which is the ability to come up with a single correct answer.

00diffracted beam intensities is to match convergent beam patterns.

01

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.