Express the following sum in terms of $n$: $\sum_{k=1}^{n} ( \frac{1}{2k-1}-\frac{1}{2k+1})$

Question

I want to express this sum in terms of $n.$

$$\sum_{k=1}^{n} \left(\frac{1}{2k-1}-\frac{1}{2k+1}\right)$$

I've read somewhere that the sum should be equal with $\frac{2n}{2n+1}$, but I don't see how I could reach that result.

Thanks for any help.

Answer 1

Define $$F_k=\frac{1}{2k-1}$$, then $T_k=F_k-F_{k+1}$. By telescoping summation we have $$T_1=F_1-F_2$$ $$T_2=F_2-F_3$$ $$T_3-F_3-F_4$$ $$..............$$ $$T_{n-1}=F_{n-1}-F_{n}$$ $$T_n=F_{n}-F_{n+1}$$ Adding them up $$\sum_{k=1}^{n} T_k=F_1-F_{n+1}=1-\frac{1}{2n+1}=\frac{2n}{2n+1}$$

Answer 2

Hint :

Write the first 8-10 terms of the sum,

$1/1-1/3+1/3-1/5+1/5-1/7...$

Do you see some terms cancel each other?

Can you make a pattern?

Answer 3

Option:

$(\dfrac{1}{2k-1}-\dfrac{1}{2k})+(\dfrac{1}{2k}-\dfrac{1}{2k+1})$