# Thread: Show how two series as the same Sum

1. ## Show how two series as the same Sum

show that the sum of the first 2n terms of the series 1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5........................is the same as the sum of the first n terms of the series 1/1*2 + 1/2*3 + 1/3*4 +1/4*5................... Do these series converge? what is the sum of the first 2n+1 terms of the first series and how to get the sum?

2. (1) Hint: 1+(-1/2+1/2)+(-1/3+1/3)+(-1/4+1/4)+(-1/5+1/5)+...+(-1/(2n) + 1/(2n))
(2) Hint: 1/(1*2) + 1/(2*3) + 1/(3*4) +1/(4*5) + ... + 1/(n*(n+1))
1/(n*(n+1)) = 1/n - 1/(n+1)

3. The first one obviously converges to 1. $1+(\frac{-1}{2}+\frac{1}{2})+(\frac{-1}{3}+\frac{1}{3})+...$ since $(\frac{-1}{2}+\frac{1}{2})=0$ they eliminate eachother.

The second is $\sum_{1}^{n}\frac{n}{n(n+1)}=\sum_{1}^{n}\frac{1}{ n}-\frac{1}{n+1}$ $=(\frac{1}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+...=1+(\frac{-1}{2}+\frac{1}{2})+(\frac{-1}{3}+\frac{1}{3})+...=1$

4. thank u guys i was mistaken 2n a 2xSn=2x(1+1/2-1/2......

5. Oh, i missed the 2n, but the argument would be very similar, use p0oints hints