Try Magic Notes and save time.Try it free
Try Magic Notes and save timeCrush your year with the magic of personalized studying.Try it free
Question

# In this exercise, determine the convergence or divergence of the series.$\displaystyle\sum_{n=1}^{\infty} \frac{n+10}{10 n+1}$

Solution

Verified

Let $a_n=\dfrac{n+10}{10n+1}$ and consider

\begin{aligned}\lim_{n\rightarrow \infty}a_n&=\lim_{n\rightarrow\infty }\dfrac{n+10}{10n+1}\\&=\lim_{n\rightarrow \infty}\dfrac{1+10/n}{10+1/n}&&(\text{Dividing both sides by }n)\\&=\dfrac{1+0}{10+0}\\&=\dfrac{1}{10}\end{aligned}

Thus we get $\lim_{n\rightarrow \infty}a_n\neq 0$ and then using $n^{th}$ term test for divergence, we get $\sum_{n=1}^{\infty}a_n=\sum_{n=1}^{\infty}\dfrac{n+10}{10n+1}$ diverges.

## Recommended textbook solutions

#### Thomas' Calculus

14th EditionISBN: 9780134438986 (3 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,144 solutions

#### Calculus: Early Transcendental Functions

4th EditionISBN: 9780618606245Bruce H. Edwards, Larson, Robert P. Hostetler
12,642 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (1 more)James Stewart
11,083 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (3 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,049 solutions