How can the Power Series for Arc Tan be Proven for Homework?

[whatlifeforme]

Homework Statement

Prove.

Homework Equations

arctan(x) = x - $\displaystyle \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9}$
for -1 < x < 1.

The Attempt at a Solution

$\displaystyle \sum^{∞}_{n=1} \frac{x^{n+2}}{n+2}$

 

[tiny-tim]

what methods have you been taught for finding the taylor series?

 

[whatlifeforme]

we skipped taylor and mclaurin series as we did not have time.

 

[tiny-tim]

then why have you been set this question??

ok, here's a "cheat" idea …

doesn't that series look a bit like an integral? ​

 

[STEMucator]

Wait. is the question to prove that arctan is equal to its power series OR to find the series for arctan?

If it's to prove it, then you may want to consider what's happening the remainder term.

 

[LCKurtz]

Homework Statement

Prove.

Homework Equations

arctan(x) = x - $\displaystyle \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9}$
for -1 < x < 1.

The Attempt at a Solution

$\displaystyle \sum^{∞}_{n=1} \frac{x^{n+2}}{n+2}$

How did you get that answer?

 

[whatlifeforme]

then why have you been set this question??

ok, here's a "cheat" idea …

doesn't that series look a bit like an integral? ​

yes, when our teach show used she used integrals in part of the problem, but I've forgotten now.

 

[FeDeX_LaTeX]

What happens when you differentiate that series?

 

[LCKurtz]

I will ask again: How did you get the answer you presented in the OP?

 

[tiny-tim]

(just got up :zzz:)

yes, when our teach show used she used integrals in part of the problem, but I've forgotten now.

ok, call the series A(x),

then (as FeDeX_LaTeX said) find dA/dx …

and then integrate! ​

 

[whatlifeforme]

tiny-tim: so differentiate each term, then go back and take the integral of each? where does that get me?

 

[Office_Shredder]

tiny-tim: so differentiate each term, then go back and take the integral of each? where does that get me?

The derivative of arctan(x) is 1/(1+x²). Can you get that Taylor series? Then try integrating both sides (remember to add an arbitrary constant which you have to deal with!)