An interesting summation

Calculus Level 3

Consider the partial sum : $\large S_n \equiv \sum_{k=1}^n (-1)^{k}$ Let $S \equiv \lim_{n \rightarrow \infty} S_n$

Which of the following affirmations are true?

S = \frac{1}{2}

S = 1

S is not even well defined!

S = -1

S = \infty

S = -\frac{1}{12}

2 solutions

Roman Frago
Feb 12, 2015

$S_n=-1$ when $n$ is odd and $S_n=0$ when $n$ is even. When $n$ $\rightarrow$ $\infty$ , we can't determine if it's odd or even.

For reference, the given partial sum is the $n^{\textrm{th}}$ partial sum of the infamous Grandi's series.

Note: This one's a bit modified since the summing starts from $k=1$ while in the original version, the sum starts from $k=0$ .

Prasun Biswas - 6 years, 3 months ago

Aman Kumar
Feb 13, 2015

Vague data We can't say about the whole summation because it is not clearly stated that whether n is even or n is odd ..........