Show that the following telescoping series is divergent. \sum_{n=1}^{\infty} (1 + \frac{1}{n})

Question:

Show that the following telescoping series is divergent.

{eq}\sum_{n=1}^{\infty} (1 + \frac{1}{n}) {/eq}

Series:

The series is termed as the telescoping series if each term is separated and then evaluated for the sum. The series will diverge if on the divergent test, the value of the limit is not equal to zero. The example is in the solution given.

Answer and Explanation:


The series that we have is :

{eq}\sum_{n=1}^{\infty} (1 + \frac{1}{n})\\ {/eq}

We will apply the series divergence test, and as per the test it says

{eq}\mathrm{If\:}\lim _{n\to \infty }a_n\ne 0\mathrm{\:then\:}\sum a_n\mathrm{\:diverges}\\ => \lim _{n\to \infty \:}\left(1+\frac{1}{n}\right)\\ \lim _{x\to a}\left[f\left(x\right)\pm g\left(x\right)\right]=\lim _{x\to a}f\left(x\right)\pm \lim _{x\to a}g\left(x\right)\\ =\lim _{n\to \infty \:}\left(1\right)+\lim _{n\to \infty \:}\left(\frac{1}{n}\right)\\ =1+0\\ =1\\ {/eq}

Thus the series is diverging in nature


Learn more about this topic:

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