# Math Help - dont understand how to approxiamte sum of a series?

1. ## dont understand how to approxiamte sum of a series?

it syas approxiamate the sum of the series to four decimal places.. do i just keep on testing out numbers and then taking the average or what?

2. What is the series?

3. What is the Summation?

4. it is ((-1)^n)n/7^n

5. $\displaystyle \sum n \left(\frac{-1}{7}\right)^n$ n=1 to infinity?

6. its (-1)^n times n/(7^n)

7. $\displaystyle\sum\limits_{n = 1}^\infty {\frac{{( - 1)^n n}}
{{7^n }}}$

8. ya thats it.. i just dont understand how to do this. do i set n to a certain value like 5?

9. EDIT

Given that $a_n\ge a_{n+1}>0~\&~(a_n)\to 0$ then $\sum\limits_{n = 1}^\infty {( - 1)^n a_n }$ converges.

If $S=\sum\limits_{n = 1}^\infty {( - 1)^n a_n }$ then $\left| {S - \sum\limits_{n = 1}^N {( - 1)^n a_n } } \right| \leqslant a_{N + 1}$.

10. Since this is an alternating series, the sum to "N+2" will always lie between the sum to "N" and "N+1". That means that the full sum will lie between the sum to N and N+1. And that means that the error will be no larger than the absolute value of the difference between those two sums which is just the absolute value of the N+ 1 term:
$\frac{N+1}{7^{N+1}}$. (I see now that is what Plato just said.)

In order to get the sum correct within 4 decimal places, that is, with error between -0.00005 and 0.00005, you must use N such that $\frac{N+1}{7^{N+1}}< 0.00005$