# Convergent Sentence Examples

- It 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. - Every
**convergent**is nearer to the value of the whole fraction than any preceding**convergent**. - Ina 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. - The 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. - It 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. - If 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. - Every even
**convergent**is greater than every odd**convergent**; every odd**convergent**is less than, and every even**convergent**greater than, any following**convergent**. - Every
**convergent**is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the**convergent**. - We may require to represent the infinite
**convergent**power series ao+alx+ a2x 2 + ... - Lambert for expressing as a continued fraction of the preceding type the quotient of two
**convergent**power series. - (iv) Each
**convergent**is nearer to the true value than any other fraction whose denominator is less than that of the**convergent**. - It 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. - But, 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. - - 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. - When 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. - Trans., 1834) in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is z z 3 25 27 J1(z) = 2 2 2.4 + 2 2.4 2.6 2 2.4 2.6 2.8 + When z is great, we may employ the semi-
**convergent**series Ji(s) = A/ (7, .- z)sin (z-17r) 1+3 8 1 ' 6 (z) 2 3.5.7.9.1.3.5 5 () 3 1 3.5.7.1 1 3 cos(z - ?r) 8 ' z (z) 3.5.7.9.11.1.3.5.7 1 5 + 8.16.24.32.40 (z - By 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 . - Descending series of the semi-
**convergent**class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x=uy, and expanding the denominator in powers of y. - The 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. - On 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. - This being so, it would be premature to disregard the
**convergent**lines of historical evidence which tell against A.D. - .; the series, however, is
**convergent**for real values of x only when x lies between +I and -1. - (-) 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. - The 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. - From 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. - The resemblance shows various grades of completeness; and the
**convergent**mimics may be themselves noxious, or edible and innocuous. - The 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). - The numerators and denominators of the successive
obey the law p n g n _ l - pn-1qn = (- O n, from which it follows at once that every**convergents****convergent**is in its lowest terms. The other principal properties of theare 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.**convergents** - The 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. - If 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 ofis even or odd, which we can take them to be as we please.**convergents** - In 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. - When 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 + ... - It is always possible to find the value of the n th
**convergent**to a recurring continued fraction. - The 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. - , 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. - In this case the sum to n terms of the series is equal to the nth
**convergent**of the fraction. - The 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. - - 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. - The 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 . - Man 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. - There is nearly always an arrangement to observe the preparation first in
**convergent**light and then in parallel polarized light. - The 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 - Its n th
**convergent**is not equal to the sum to n terms of the series. - (ii) Each
**convergent**is a fraction in its lowest terms. - A 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.