# 009C Sample Midterm 1

This is a sample, and is meant to represent the material usually covered in Math 9C for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

## Problem 1

Does the following sequence converge or diverge?

If the sequence converges, also find the limit of the sequence.

${\displaystyle a_{n}={\frac {\ln n}{n}}}$

## Problem 2

Consider the infinite series  ${\displaystyle \sum _{n=2}^{\infty }2{\bigg (}{\frac {1}{2^{n}}}-{\frac {1}{2^{n+1}}}{\bigg )}.}$

(a) Find an expression for the  ${\displaystyle n}$th partial sum  ${\displaystyle s_{n}}$  of the series.

(b) Compute  ${\displaystyle \lim _{n\rightarrow \infty }s_{n}.}$

## Problem 3

Determine whether the following series converges absolutely,

conditionally or whether it diverges.

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}}$

## Problem 4

Determine the convergence or divergence of the following series.

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}3^{n}}}}$
(a)  ${\displaystyle \sum _{n=0}^{\infty }{\sqrt {n}}x^{n}}$
(b)  ${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{\frac {(x-3)^{n}}{2n+1}}}$