calculus - Find a simple formula for the partial sum

20. The series $\sum_{n=1}^\infty\frac1{4n^2-1}$ telescopes. Find a simple formula for the $k$th partial sum $S_k$, and use it to determine whether the series converges. If it converges, find its value.

I've mostly tried to intuit my way through. Calculating the partial sums to several degrees by varying $n$, I got

$$\frac13+\frac1{15}+\frac1{35}+\frac1{63}+\frac1{99}+\cdots$$ and I can see that it is converging to $\frac12$.

I computed the first few multiplicative differences between the values and saw that $\frac13$ was multiplied by $\frac15$ to get $\frac1{15}$, $\frac1{15}$ by $\frac37$, $\frac1{35}$ by $\frac7{11}$.

I see that both the denominator and the numerator of the multiplier is increasing by two each time. So I see that it is certainly converging. But I cannot quite put the pieces together on the formula. Any help?

Answer

Basically the telescoping technique is like this: You have a sequence $a_1, a_2 , \cdots , a_n \cdots$. You find a sequence $b_1, b_2 , \cdots , b_n \cdots$ such that $a_k = b_k - b_{k+1}$. So summing $a_k$ would be like summing

$$\sum_{k=1}^n a_k = \sum_{k=1}^n( b_k - b_{k+1}) = b_1 - b_2 + b_2 - b_3 + b_3 - b_4\cdots + b_n - b_{n+1}.$$

You can see here that the terms $- b_2 + b_2$, $- b_3+ b_3$ etc. vanish when you sum. So you only have to pick up the first and the last term:

$$\sum_{k=1}^n a_k = \sum_{k=1}^n b_k - b_{k+1} = b_1 - b_{n+1}.$$

So what you need to do here is to find $b_n$ such that $\frac{1}{4n^2 - 1} = b_n - b_{n+1}.$