This is a question I came across while studying on Khan Academy:
Find the sum of the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\left( \frac { n+2 }{ { n }^{ 5 } } \right) } $ correct to three decimal places.
I solved this by calculating ${ S }_{ 1 },{ S }_{ 2 },{ S }_{ 3 },…\\ $ until I had a close enough value (bad method, I know).
Anyway, here is the solution steps given in the answer:
Note that this is a convergent alternating series. The Alternating Series Estimation Theorem states that the error bound in the sum of the first $n$ terms is the absolute value of the first omitted term; i.e., $\left| { a }_{ n+1 } \right| $. To obtain three-digit accuracy, we need to find $n$ such that $\left| { a }_{ n+1 } \right| <\ 0.0005$, or equivalently $\frac { n+3 }{ { (n+1) }^{ 5 } } <0.0005$.
When $n=6$, we have $\left| { a }_{ n+1 } \right| =\frac { n+3 }{ { (n+1) }^{ 5 } } =\frac { 9 }{ 16807 } \approx .000535$; that is not accurate enough.
When $n=7$, we have $\left| { a }_{ n+1 } \right| =\frac { n+3 }{ { (n+1) }^{ 5 } } =\frac { 10 }{ 32768 } \approx .000305$; that will give a final result within $0.0005$.
Thus we need to find the sum of the first seven terms of the series.
$$\sum _{ n=1 }^{ 7 }{ { (-1) }^{ n+1 }\left( \frac { n+2 }{ { n }^{ 5 } } \right) \approx 2.8914 } $$
Hence, $2.891$ is accurate to three decimal places.
What I don't understand about this answer is if the estimate is $2.8914$ and the error bound is $0.000305$, I think the actual sum can be close to $2.8914+0.000305\approx 2.891705\approx 2.892$. Then the result above is not correct.
Does this answer take into account the fact that the $(7+1)th$ term is negative? Or is there something about rounding and error bounds that I don't understand?
Thank you.
Best Answer
I would read the question the same way you did-find a three decimal value that you know the true sum of the series rounds to. We know from the alternating series theorem that the truncation error is smaller than the first neglected term in absolute value and of the same sign. There is no truncation error that we can guarantee will be small enough, because we could be poised right on the rounding boundary, with the computed answer something like $2.8915$. Here, because after seven terms we have $2.8914$ and we know the eighth term is negative, the correct answer is in the range $(2.8911,2.8914)$ and this entire range rounds to $2.891$