Does the Series Converge? A Comparison Test Approach

[chwala]

Homework Statement

determine whether the series below converges.
$\sum_{n=1}^\infty 2^n.n+1,√(n^4+4^n.n^3)$

Homework Equations

The Attempt at a Solution

 

[BvU]

Useful things, brackets !

$${\sum_{n=1}^\infty {(2^n.n+1)}\over \sqrt {n^4+4^n.n^3}} \quad \rm ?$$

And 'whether the series ...' what exactly ?

How about your own attempt at solution ? What instruments do you intend to usse ?

 

[chwala]

Useful things, brackets !

$${\sum_{n=1}^\infty {(2^n.n+1)}\over \sqrt {n^4+4^n.n^3}} \quad \rm ?$$

And 'whether the series ...' what exactly ?

How about your own attempt at solution ? What instruments do you intend to usse ?

sorry unable to post in latex form, i will attempt and share.

 

[chwala]

$2^n+1/√n^3(n+4^n)$

 

[BvU]

Hello Chwala,

Let me get this right. You mean $$
2^n+ {1\over \sqrt {n^3(n+4^n)} }\quad \rm ? $$ No ? I thought so.

 

[chwala]

Sorry i meant $(2^n. n+1)/√{n^3(n+4^n)}$
= $(2^n. n+1)^2/{n^3(n+4^n)}$

 

[BvU]

The equals sign can't be right. Generally $y \ne y^2$ Which one of the two ?
And what happened to the $\Sigma$ ?
And what is it we want to know about whatever the correct expression turns out to be ?

 

[chwala]

agreed we just deal with y me thinks we have to try express it in geometric from and check for |r| since n=1, we have to use the n-1 form..

 

[BvU]

We still don't know where we are (what happened to the square root ?")

And then you might also reveal where we want to go. You still haven't told us.

My lunch break is almost over and all we did was find out what you want. Perhaps you want to check your postings before shooting them off. Try to be as dumb and precise as the worst of your readers. (me ? )

 

[chwala]

hahahhahaha i will re post again only challenge is the latex...let us consider $(2^n. n+1)/√{n^3(n+4^n)}$ as my attempt and we forget about the other step.

 

[BvU]

Ok square root is back. Could you please please please stop writing a/bc -- which is equal to a/b times c -- when you really mean a/(bc). It is unforgivable in these situations. Use brackets. They are cheap (in fact, they come for free). Please confirm reception of this advice .

Homework Statement

determine whether the series below converges $$ \sum_{n=1}^{\infty} { n \;2^n +1 \over \sqrt {n^3\;(n+4^n)} } $$​

Are we there yet ?

 

[Ray Vickson]

hahahhahaha i will re post again only challenge is the latex...let us consider $(2^n. n+1)/√{n^3(n+4^n)}$ as my attempt and we forget about the other step.

You need not use LaTeX, and you probably should not until you can use it correctly! It can all be done reasonable clearly in plain text if you use parentheses, like this: (2^n * n+1)/sqrt[n^3 *(n + 4^n)].

That said, I urge you to take the time to learn how to use LaTeX properly; you can right-click on displayed expressions others have written, and ask for the expression to be displayed as tex commands. Remember also: you need to tell the PF server that "latex starts here" and "latex ends here". For an in-line expression just use # # (with no space between the # and the #) at both the start and end of your expression, to get $\text{some expression}\cdots$. For a "displayed" version, use "[ tex ]" (but with no spaces) at the start and "[ /tex ]" (no spaces) at the end, to get
\text{some expression} \cdots

 

[Mark44]

For a "displayed" version, use "[ tex ]" (but with no spaces) at the start and "[ /tex ]" (no spaces) at the end

My preference for standalone LaTeX, as it entails less typing is to use $$ at the start and $$ at the end. For inline expressions, I use ## at both ends, like you said.

 

[micromass]

Soo, now we finally know the problem! So what did you already try to solve it. There are various tests that you can use, so did you try them?

 

[chwala]

I looked into this one can use ratio test or a direct comparison test with a similar series...

 

[BvU]

My preference for standalone LaTeX, as it entails less typing is to use $$ at the start and $$ at the end. For inline expressions, I use ## at both ends, like you said.

How do you type dollar dollar $$ without $$ ending up in $\LaTeX$ ?

 

[BvU]

I looked into this one can use ratio test or a direct comparison test with a similar series...

And what is the result ?

 

[Mark44]

How do you type dollar dollar $$ without $$ ending up in $\LaTeX$ ?

By changing the color of one of the $ symbols at each end.
$$\frac{x - 1}{x + 3}$$
In the expression above, the 2nd and 4th $ symbols have color attributes.

 

[micromass]

I looked into this one can use ratio test or a direct comparison test with a similar series...

So what did you get with the ratio test? With what series did you compare?

 

[chwala]

Bvu i will post the results soon...sorry i am on holiday am enjoying the beach of mombasa, kenya

 

[chwala]

sorry back from holiday... The problem was to establish convergence of the series $\sum_{n=1}^{\infty} { 2^n \;n+1 \over \sqrt {n^3\;(n+4^n)} }$
The little knowledge i have on this is for example to use the comparison test and compare it with a similar looking series...i will attempt and post accordingly...