Find the sum of the following series

In summary, the person is asking for a power series representation for a function that takes a negative number as an argument. They are also asking for help verifying convergence/divergence.
  • #1
verd
146
0
Hi,

I'm having a bit of difficulty with the following two problems:

The first asks to find the sum of the following series:
[tex]\sum\limits_{n = 1}^\infty {\frac{2}{n^2+4n+3}} [/tex]

I easily understand geometric series and how to find sums with those, but this seems a bit complicated. Factoring and pulling the 2 out doesn't seem to really help any. ...Any suggestions?


And this question:

Find a power series representation for the following:
[tex]g(x)=\arctan{\frac{x}{2}}[/tex]

I understand the simple 1/4+x^2 kind of stuff, but this I don't even know how to begin.


Any suggestions??


Thanks!
 
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  • #2
...Telescoping series for #1??
 
  • #3
verd said:
[tex]\sum\limits_{n = 1}^\infty {\frac{2}{n^2+4n+3}} [/tex]

I would suggest checking to see if partial fraction decomposition on the summand makes the problem easier to get at.

After you break it apart you should get something like

[tex]
\sum\limits_{n = 1}^\infty {\frac{1}{n+1}} - \sum\limits_{n = 1}^\infty {\frac{1}{n+3}}
[/tex]

Then notice that this telescoping series has only two terms that don't get canceled out. Add those two terms to get the sum.
 
Last edited:
  • #4
Wouldn't that work for testing convergence/divergence?

I need to find the sum... I wasn't aware that by partial fractions/integrals you could find sums... Is that possible? If so, what are the conditions?
 
  • #5
verd said:
Find a power series representation for the following:
[tex]g(x)=\arctan{\frac{x}{2}}[/tex]

What's the derivative of arctan(x)?

Can you find a power series representation for that? Can you integrate term by term? :smile:
 
  • #6
verd said:
Wouldn't that work for testing convergence/divergence?

I need to find the sum... I wasn't aware that by partial fractions/integrals you could find sums... Is that possible? If so, what are the conditions?

That works fine for the sum. The partial faction decomp isn't just for integrals. Try adding the two sums together and see that you get back your original summand. Once you're convinced they are in fact the same, then notice that every n+2 for the first summand is the additive inverse of the nth term of the second summand. You should notice that only two terms are not canceled out in this manner. Add those two terms to get the sum.
 
  • #7
Woo, I figured it out right before you replied! Haha, thanks, you verified that what I was doing was correct! Thanks!

Anyone have any ideas on that second one?
 
  • #8
verd said:
Anyone have any ideas on that second one?

Look three post up from this one...
 
  • #9
Can someone tell me if this is correct for the second problem posted?

[tex]\sum\limits_{n = 1}^\infty {\frac{(-1)^{n}x^{2n-1}}{(2n-1)4^{n-1}}} [/tex]
 
  • #10
verd said:
Can someone tell me if this is correct for the second problem posted?

[tex]\sum\limits_{n = 1}^\infty {\frac{(-1)^{n}x^{2n-1}}{(2n-1)4^{n-1}}} [/tex]

The easiest way (non-rigorous, but fast) to verify your answer is to work out around five to six terms with some "not so nice" value for x (like 0.23) and confirm that the sum is tending to the correct value for arctan(0.23/2). Quite fast with a scientific calc and the Memory+ function.
 

FAQ: Find the sum of the following series

What is the purpose of finding the sum of a series?

The purpose of finding the sum of a series is to determine the total value of all the numbers in the series combined. This can be useful in various mathematical and scientific calculations.

How do you find the sum of a series?

To find the sum of a series, you add up all the numbers in the series. This can be done manually by writing out each number and adding them together, or by using a mathematical formula specific to the type of series.

What are some common types of series?

Some common types of series include arithmetic series, geometric series, and power series. Each type has a specific formula for finding the sum of the series.

Can all series be summed up?

No, not all series can be summed up. Some series, such as divergent series, do not have a finite sum and therefore cannot be summed up.

Why is it important to understand how to find the sum of a series?

Understanding how to find the sum of a series is important in many fields of science and mathematics. It can be used in financial calculations, statistical analysis, and even in physics and engineering to determine total values and make predictions.

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