Partial Sum For a certain kind of series.

Hi everyone! Today we were asked to say if this particular series converges or diverges : $\displaystyle 1+\frac{10}{4}+\frac{10}{9}+\frac{10}{16}...$. The first thing I though about is to find if the partial sum converges in to something, but the thing is, I can't find a formula to define the nth partial sum of the particular series. It looks like the denominator is just $\displaystyle n^2$ but I can't understand why the first summand would be only one where as all the other summands seem to be multiplied by 10. Anyone got any idea on how to find the formula for the partial sum or at least have another method to prove its convergence or divergence? Thanks everyone in advance!(Smile)

Re: Partial Sum For a certain kind of series.

Quote:

Originally Posted by

**EliteAndoy** Hi everyone! Today we were asked to say if this particular series converges or diverges : $\displaystyle 1+\frac{10}{4}+\frac{10}{9}+\frac{10}{16}...$. The first thing I though about is to find if the partial sum converges in to something, but the thing is, I can't find a formula to define the nth partial sum of the particular series. It looks like the denominator is just $\displaystyle n^2$ but I can't understand why the first summand would be only one where as all the other summands seem to be multiplied by 10. Anyone got any idea on how to find the formula for the partial sum or at least have another method to prove its convergence or divergence? Thanks everyone in advance!(Smile)

You can write it as

$\displaystyle \displaystyle \begin{align*} 1 + \frac{10}{4} + \frac{10}{9} + \frac{10}{16} + \dots &= 1 - 10 + \frac{10}{1} + \frac{10}{4} + \frac{10}{9} + \frac{10}{16} + \dots \\ &= -9 + 10\sum_{k = 1}^{\infty}{\frac{1}{k^2}} \\ &= -9 + 10 \left( \frac{\pi ^2}{6} \right) \\ &= \frac{5 \pi ^2 - 27}{3} \end{align*}$

Re: Partial Sum For a certain kind of series.

Dang, never thought of subtracting 9 from the harmonic sequence. I'd certainly remember this one for other upcoming problems. Thanks a lot man!

Re: Partial Sum For a certain kind of series.

Prove It not only proved that it converges, but actually came up with the sum!

But the question only asks for convergence or divergence, and it is a p-series, so you can state immediately that it converges since p=2. Changes to the initial terms do not affect the convergence, and neither does multiplication by a constant.

- Hollywood