- #1

frixis

- 19

- 0

**Series and Sequence Convergence... final on monday please help!**

## Homework Statement

determine whether the sequence converges or diverges and find its limits

1[itex](1+\frac{1}{n})^n[/itex]

2 cos[tex]\pi[/tex]n

the overbar indicates that the digits underneat repeat indefinitely. Express as a series and find the rational number(since i can't find the latex for overbar, the brackets represent the overbar)

0.(23)

3.2(394)

show divergence or convergence by the nth term text

[itex]\displaystyle\sum_{n=2}^{\infty} \frac{1}{n\sqrt{{n^2}-1}}[/itex]

use a basic comparision test to determine convergence

[itex]\displaystyle\sum_{n=1}^{\infty} \frac{\arctan n}{n}[/itex]

determine convergence

1[itex]\displaystyle\sum_{n=1}^{\infty} \frac{2n+n^2}{n^3+1}[/itex]

2[itex]\displaystyle\sum_{n=1}^{\infty} \frac{n^2+2^n}{n+3^n}[/itex]

3[itex]\displaystyle\sum_{n=1}^{\infty} \frac{\mathrm{ln} n}{n^3}[/itex]

4 [tex]\displaystyle\sum_{n=2}^{\infty} {\frac{1}{{n^3}{\sqrt[3]{\mathrm{ln}n}}}[/tex]

5[itex]\displaystyle\sum_{n=1}^{\infty} n{\tan{\frac{1}{n}}[/itex]

6[itex]\displaystyle\sum_{n=1}^{\infty} {(1+\frac{1}{n})}^n[/itex]

7 [itex]1+\frac{1\cdot 3}{2!}+\frac{1\cdot 3\cdot 5}{3!}+...+\frac{1.3.5...(2n-1)}{n!}+...[/itex]

find every real no k for which the series converges

[itex]\displaystyle\sum_{n=2}^{\infty} \frac{1}{n^k{\mathrm{ln}n}}[/itex]

use the proof of the integral test to show that, for every positive int n >1

[itex]{\mathrm{ln}(n+1)}{<1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}<1+{\mathrm{ln}n}[/itex]

determine wheter the series is absolotely convergent, conditionally convergent or divergent.

[itex]\displaystyle\sum_{n=1}^{\infty} {(-1)}^n \frac{\cospin}{n}[/itex]

show that the alternating series converges for ever positive integer k

[itex]\displaystyle\sum_{n=1}^{\infty} {(-1)}^n \frac{{(\mathrm{ln}n)}^k}{n}[/itex]

find the interval of convergence of the power series

1 [itex]\displaystyle\sum_{n=1}^{\infty} {(-1)}^n \frac{x^n}{\sqrt{n}}[/itex] can do the whole getting to the interval just can't decide whether or not it converges or diverges at -1 and 1 which are the intervals endpoints

2 [itex]\displaystyle\sum_{n=0}^{\infty} {\frac{n!}{100^n} x^n}[/itex] issues with endpoints again

3 [itex]\displaystyle\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(-4)^n}[/itex]

find the radius of convergence of the power series for positive integers c and d

[itex]\displaystyle\sum_{n=0}^{\infty} {\frac{{(n+c)}!}{n!{(n+d)}!} x^n[/itex]

if the interval of convergence of [itex]\displaystyle\sum {a_n}{x^n}[/itex] is (-r,r], prove that the series is conditionally convergent at r

## The Attempt at a Solution

um.. okay so i have to put down the attempts to all the questions? its a culmination of all issues...

for the decimals ones... i did the whole sum = a/1-r thing... for some reason i was doing something wrong...

for all convergence ones i always tried the ratio test or the root test... but i still had issues with the convergence... specially when ln comes in... i don't know y...

and the last one which is the most recent... i deduced that the ratio of the n+1 term and the n term is 1/r but then i tried putting that in the equation and i was pretty lost... because u'd get 1 and... yeah... okay...

so if i need to post all my attempts please let me know...

Last edited: