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158. Tests for Convergence. From what we have already seen, an obvious necessary test for convergence is that the value of an individual term shall approach zero as a limit as we go indefinitely further out in the series; i. e., that

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Unless, in addition, the signs of the terms of the series alternate, this is not sufficient for convergence, and even then does not ensure absolute convergence.

Consider the test-ratio, or ratio of convergence,

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This commonly varies as we take terms at different places, and we shall show that if, as we go further and further out in the series, it approaches a limit numerically less than 1, the series will converge absolutely; if it approaches a limit numerically greater than 1, or increases without limit, the series will diverge; if it approaches 1 or 1 as a limit, the series may either converge or diverge.

159. Comparison-Test.-First, if we have two series,

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the second of which is known to be convergent, and if each term of the first series is numerically less than the corresponding term of the second series, then it is evident that the first series is convergent.

For instance, since

1 + 1 + ( 1 )2 + ( 1 ) 3 + ( 1 ) * + . . . .

is convergent, it is evident that

1+1 (1) + (1)2 + 1 ( 1 ) 3 + b ( 1 ) * +....

is also convergent.

The test by means of the ratio of convergence is derived by comparing any given series with a geometric progression.

Given any series

T1, T2, T3, T4,

(1)

and supposing that from the kth term on, the ratio of convergence is numerically less than some constant number a which itself is numerically less than 1 (i. e., r,<a<1), then the series (1) is convergent, for

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is numerically less than the corresponding term of the series

aTk, a2Tk, a3Tk, a⭑Tk, ....

(3)

The series (3) is a geometric progression with common ratio a numerically less than 1, and so is convergent; hence the series (2) is convergent, and also the series (1), from which the series (2) was obtained by the omission of a definite number of terms.

The second part of the test can be proved by using in a similar way the evident principle that if every term of a given series is numerically greater than the corresponding term of a divergent series, the given series is also divergent.

When [n]n==±1, neither of these comparisons will apply; there are in fact both convergent and divergent series for which this limit is numerically equal to unity.

160. Test by the Limit of the Ratio of Convergence.-This test is formulated for reference as follows:

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L, and L numerically <1, series converges.

n=8

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L', and I' numerically >1, series diverges.

=±1, no test.

Examples.

Test the following series for convergence, giving the region of

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161. The Exponential Theorem.-The exponential function, a, in which a is the variable and a is any constant, is expanded into an ascending power-series as follows:

Assume

a = b+cx+dx2 + ex3 +

(1)

As this is an identity, x can be replaced by any number; write 2x for x:

a2=b+2cx+4dx2+8ex3 +.....

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(2).

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and so on.

+2dex + 2dfx® + ·

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We have so far been able to determine the desired coefficients only in terms of one of their own number, c, which is itself still undetermined.

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The value of this numerical series is represented by e; the computation of e to five decimals is as follows:

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and the value of c is definitely determined.

The use of the variable as an exponent has introduced a new function, of the utmost importance in mechanics and physics, the treatment of which is impossible without the use of infinite series.

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