Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series - Calculus | Socratic (2024)

FAQs

What is the Leibnitz test for convergence of an alternating series? ›

Alternating series, Leibnitz's alternating series test, Series of positive terms. A series the terms of which are alternately positive and negative is called the alternating series. An alternating series converge if the absolute values of its terms decrease monotonically to zero as n tends to infinity.

In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit.

What do you do if the Alternating Series Test fails? In most cases, an alternation series ∞∑n=0(−1)nbn fails Alternating Series Test by violating limn→∞bn=0 . If that is the case, you may conclude that the series diverges by Divergence (Nth Term) Test.

The leibniz rule states that if two functions f(x) and g(x) are differentiable n times individually, then their product f(x). g(x) is also differentiable n times. The leibniz rule is (f(x). g(x))n=∑nCrf(n−r)(x). gr(x)

Convergence. Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. Calculating π to 10 correct decimal places using direct summation of the series requires precisely five billion terms because 4/2k + 1 < 10⁻¹⁰ for k > 2 × 10¹⁰ − 1/2 (one needs to apply Calabrese error bound).

For an alternating series ∞∑n=1(−1)n+1bn, if bk+1≤bk for all k and bk→0 as k→∞, the alternating series converges.

The geometric series convergence formula is a 1 − r if |r| < 1, where a is the first term and r is the common ratio, i.e., the number that each term is multiplied by in order to produce the next term. Some people refer to it as a formula, but it is both a formula and a test.

The alternating series test doesn't help to prove absolute converges. You need to show that the series of absolute values ∑∞n=1|an| converges.

Therefore, the behavior of the infinite series can be determined by looking at the behavior of the sequence of partial sums Sk. If the sequence of partial sums Sk converges, we say that the infinite series converges, and its sum is given by limk→∞Sk. If the sequence Sk diverges, we say the infinite series diverges.

What is the convergence condition for infinite series? ›

There is a simple test for determining whether a geometric series converges or diverges; if −1<r<1, then the infinite series will converge. If r lies outside this interval, then the infinite series will diverge. Test for convergence: If −1<r<1, then the infinite geometric series converges.

The sum of a convergent infinite geometric series is ∑ k = 1 ∞ a r k − 1 = a 1 − r where a is the first term and r is the common ratio. In a geometric series, r is also the number being raised to an exponent in the sigma notation formula.

Alternating Series and Leibniz's Test Let a1,a2,a3,... be a sequence of positive numbers. A series of the form a1 − a2 + a3 − a4 + a5 − a6 + ... is said to be alternating because of the alternating sign pattern.

If property 3 is respected but property 1 and/or property 2 do not hold, then the alternating series test is inconclusive. It is easy to exhibit a divergent series that satisfies properties 1 and 3 but does not satisfy property 2.

We cannot remove a finite number of terms to make decreasing, therefore we cannot apply the alternating series test. Keep in mind that this does not mean we conclude the series diverges; in fact, it does converge. We are just unable to conclude this based on the alternating series test.

Basically, the Leibnitz theorem is used to generalise the product rule of differentiation. It states that if there are two functions let them be a(x) and b(x) and if they both are differentiable individually, then their product a(x). b(x) is also n times differentiable.

The Alternating Series Test.

Given an alternating series , ∑ ( − 1 ) k a k , if the sequence of positive terms decreases to 0 as , k → ∞ , then the alternating series converges. Note that if the limit of the sequence is not 0, then the alternating series diverges.

In other words, if the absolute values of the terms of an alternating series are non-increasing and converge to zero, the series converges. This is easy to test; we like alternating series.

Strategy to test series

If a series is a p-series, with terms 1np, we know it converges if p>1 and diverges otherwise. If a series is a geometric series, with terms arn, we know it converges if |r|<1 and diverges otherwise. In addition, if it converges and the series starts with n=0 we know its value is a1−r.