MHB Jacob's question at Yahoo Answers (Alternating series approximation)

AI Thread Summary
The discussion focuses on approximating the sum of the series ∑ (-1)^n/(5^n*n!) from n=1 to infinity, aiming for four decimal places. The Leibniz criterion is highlighted, indicating that the series converges if the terms decrease monotonically to zero. The error in the approximation is bounded by the next term, leading to the condition that a_{k+1} must be less than 0.0001. The smallest integer k that satisfies this condition is found to be 4, allowing for the calculation of the partial sum S_k. The final approximation of the series is derived from summing the first four terms.
Fernando Revilla
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Here is the question:

Approximate the sum of the series correct to four decimal places.

∑ (-1)^n/(5^n*n!)
n=1

Here is a link to the question:

Approx. series help please? - Yahoo! AnswersI have posted a link there to this topic so the OP can find my response.
 
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Hello Jacob,

The Leibniz criterion for the alternating series $\sum_{n=1}^{\infty} (-1)^{n-1}a_n$ says: if $a_n$ decrease monotonically to $0$ as a sequence of positive numbers when $n$ approaches infinity, i.e. then the alternating series converges. Moreover, let $S$ denote the sum of the series, then the partial sum $S_k =\sum_{n=1}^k (-1)^{n-1}a_n$ approximates $S$ with error bounded by the next omitted term: $\left | S_k - S \right|\leq a_{k+1}$. Then, $$a_{k+1}=\dfrac{1}{5^{k+1}(k+1)!}<0.0001 \Leftrightarrow 10\;000<5^{k+1}(k+1)!\qquad (*)$$ The smallest positive integer satisfying $(*)$ is $k=4$ so, $$S\approx \sum _{n=1}^4\frac{(-1)^n}{5^nn!}=\ldots$$ which approximate the sum of the series to four decimal places.
 
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