Show that the series is convergent. How many terms of the...

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

$ \displaystyle \sum_{n = 1}^{\infty} \left( - \frac {1}{n} \right )^n (|error| < 0.00005) $

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Alternating Series Test

The Alternating Series Test is used to determine the convergence of series whose terms alternate in sign. It states that if the absolute values of the terms are monotonically decreasing and approach zero, then the series converges. This test is crucial for understanding the convergence behavior of series like the one given, where the pattern of sign alternation can be exploited to show convergence even when standard convergence tests for non?alternating series might not apply.

Error Estimation in Alternating Series

In the context of alternating series, error estimation is governed by the Alternating Series Remainder Theorem. This theorem implies that the absolute error when approximating the sum by the partial sum is less than or equal to the absolute value of the first omitted term. This concept allows us to determine how many terms are needed to achieve a desired accuracy, by setting the absolute value of the next term less than the given error tolerance.

Transcript

00:01 First, let's show that this series is conversion.

00:04 So let's use the alternating series test.

00:13 So in this case, we should take bn to just be 1 over n to the nth power.

00:19 And this easily satisfies all the conditions.

00:23 On one hand, we can see that this is always positive because numerator and denominator are positive.

00:30 The limit of bn, so this is the second condition, is zero because.

00:38 The denominator goes to infinity, numerator is just one.

00:43 And lastly, bn plus 1 is less than or equal to bn since if we just replace n with n plus 1, the denominator gets larger, but as a whole, the fraction gets smaller.

01:00 So that's true too for all n.

01:03 So this series converges by the alternating series theorem.

01:10 Bn satisfies all three hypotheses.

01:13 And that's all that's needed.

01:16 Now there's a second part of the question, so let's maybe switch color here to blue.

01:20 How many terms of the series do we need to add to ensure a sum of the indicated accuracy? so here we just would like to know how big should n be so that if we approximate the sum using this s n, that the approximate error is less than four zeros followed by a five after the decimal.

01:49 So here we'll use the alternating series estimation theorem.

01:57 So this is also an 11 .5...

Please give Ace some feedback