Identify the pattern of terms

Question

Here is an exam question of the unit infinite series and series of functions.

Identify the pattern of terms and determine whether the following series is convergent

$$1 -1+\frac{1}{2} -\frac{1}{2} + \frac{1}{3} - \frac{1}{3} + \frac{1}{4} - \frac{1}{4} \cdots$$

Am I supposed to find the $n^{th}$ term?

Are there any method of finding a simple general term for the series?

Answer 1

The sum of this series is $0$.

For example for $n\geq3$ the general term is $$\frac{(-1)^{n+1}}{2\left[\frac{n-1}{2}\right]}$$

For an even $n$ we have $S_n=0$, while for an odd $n$ we have $S_n=\frac{1}{\frac{n+1}{2}}\rightarrow0.$

Answer 2

The sum of the first $n$ terms of your series is$$\begin{cases}0&\text{ if }n\text{ is even}\\\frac1k&\text{ if }n=2k-1\text{ for some }k\in\mathbb{N}.\end{cases}$$Therefore, the sum of your series, which is, by definition, the limit of the partial sums, is $0$.

Answer 3

Consider the partial sums.

We have that $S_N = 0$ when $N=2k$ and $S_N = \frac 1{k}$ when $N = 2k-1$. Now obviously we have that $S_N \to 0$ and so the series converges to $0$.

Answer 4

The sequence of partial sum of the series, is $$ 1,0,1/2,0,1/3,0,....$$ Which is simply $$ S n = \frac{1-(-1)^n}{n+1}$$ Therefore the series converges to $0.$