# Series and Sequence Convergence final on monday

• frixis
In summary: You have been told to post your work. If you are not going to do it, then don't post and say you will help him by private messaging him. To the OP: Can you post your actual work up to this point? What have you tried? What are you having trouble with? Please explain in as much detail as you can, so that we can help you effectively.In summary, the conversation was about determining convergence and divergence of different series using tests such as the ratio test, root test, and comparison test. The conversation also included finding the limits and intervals of convergence for power series. The student mentioned having trouble with using the ratio test and root test, especially when dealing with logarithms. They also mentioned difficulty
frixis

## Homework Statement

determine whether the sequence converges or diverges and find its limits
1$(1+\frac{1}{n})^n$
2 cos$$\pi$$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
$\displaystyle\sum_{n=2}^{\infty} \frac{1}{n\sqrt{{n^2}-1}}$
use a basic comparision test to determine convergence
$\displaystyle\sum_{n=1}^{\infty} \frac{\arctan n}{n}$
determine convergence
1$\displaystyle\sum_{n=1}^{\infty} \frac{2n+n^2}{n^3+1}$
2$\displaystyle\sum_{n=1}^{\infty} \frac{n^2+2^n}{n+3^n}$
3$\displaystyle\sum_{n=1}^{\infty} \frac{\mathrm{ln} n}{n^3}$
4 $$\displaystyle\sum_{n=2}^{\infty} {\frac{1}{{n^3}{\sqrt[3]{\mathrm{ln}n}}}$$
5$\displaystyle\sum_{n=1}^{\infty} n{\tan{\frac{1}{n}}$
6$\displaystyle\sum_{n=1}^{\infty} {(1+\frac{1}{n})}^n$
7 $1+\frac{1\cdot 3}{2!}+\frac{1\cdot 3\cdot 5}{3!}+...+\frac{1.3.5...(2n-1)}{n!}+...$
find every real no k for which the series converges
$\displaystyle\sum_{n=2}^{\infty} \frac{1}{n^k{\mathrm{ln}n}}$

use the proof of the integral test to show that, for every positive int n >1
${\mathrm{ln}(n+1)}{<1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}<1+{\mathrm{ln}n}$

determine wheter the series is absolotely convergent, conditionally convergent or divergent.
$\displaystyle\sum_{n=1}^{\infty} {(-1)}^n \frac{\cospin}{n}$

show that the alternating series converges for ever positive integer k
$\displaystyle\sum_{n=1}^{\infty} {(-1)}^n \frac{{(\mathrm{ln}n)}^k}{n}$

find the interval of convergence of the power series
1 $\displaystyle\sum_{n=1}^{\infty} {(-1)}^n \frac{x^n}{\sqrt{n}}$ 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 $\displaystyle\sum_{n=0}^{\infty} {\frac{n!}{100^n} x^n}$ issues with endpoints again
3 $\displaystyle\sum_{n=0}^{\infty} \frac{x^{2n+1}}{(-4)^n}$

find the radius of convergence of the power series for positive integers c and d
$\displaystyle\sum_{n=0}^{\infty} {\frac{{(n+c)}!}{n!{(n+d)}!} x^n$

if the interval of convergence of $\displaystyle\sum {a_n}{x^n}$ 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:

You have to post attempts to every single question. I suggest you give them a harder try, reduce the list of questions you need help with because you seem to have given us the entire list of questions, not the specific ones you need help with.

For every question you show us a decent attempt at, we will give you help on. So try to reduce the list first.

perhaps just show us your attempt on one or two of these and then we may be able to point out something that is confusing you or gaps in your knowledge.

i'll work them out and if you still need help just message me...=)

nuclearrape66 said:
i'll work them out and if you still need help just message me...=)

No, that is not in accordance with forum rules. If he wants help, he is going to post his work on this thread, not message you to get some solutions.

private messaging has nothign to do with forum posting...understand these rules...

So you are going to hand out answers to someone who obviously needs help on how to do them? Wow, that helps him learn.

Sorry, but I am seriously not going to watch you stuff up this guys understanding even further, he's already behind in his class and your going to make it worse.

## What is the difference between a series and a sequence?

A series is the sum of the terms in a sequence, while a sequence is a list of numbers in a specific order.

## What does it mean for a series or sequence to converge?

A series or sequence converges when its terms approach a finite value or limit as the number of terms increases.

## How do you determine if a series or sequence converges?

There are several tests, such as the comparison test and the ratio test, that can be used to determine if a series or sequence converges. It is important to check for conditions such as the terms approaching zero and the terms decreasing in size.

## What is the significance of convergence in mathematics?

Convergence is an important concept in mathematics as it allows us to determine the behavior and limit of a series or sequence. It also has many applications in various fields, such as physics, engineering, and economics.

## What are some real-world examples of series and sequence convergence?

An example of series convergence is the calculation of compound interest, where the terms of the series represent the interest earned each period. An example of sequence convergence is the measurement of a swinging pendulum, where the distance traveled by the pendulum decreases with each swing until it eventually stops at its equilibrium point.

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