Need some help with a power series question

[rir0302]

"Find the power series representation of arctan(x²) and then find the nth term of the series."

The hint the teacher gave is that 1/(1+x²) is the derivative of arctan.
But I have trouble with the power series concept so I need some help with how do go about doing this problem.

I assume the series would be written in this format:


Thanks in advance!

 

[firemath]

What have you done so far?

 

[topsquark]

"Find the power series representation of arctan(x²) and then find the nth term of the series."

The hint the teacher gave is that 1/(1+x²) is the derivative of arctan.
But I have trouble with the power series concept so I need some help with how do go about doing this problem.

I assume the series would be written in this format:
View attachment 14764

Thanks in advance!

Hint: Try looking at a Taylor series about x = 0. (We can worry about if it converges later.)

-Dan

 

[pka]

"Find the power series representation of arctan(x²) and then find the nth term of the series."

If \(\displaystyle |x|<1\) it is well known that \(\displaystyle F(x) = \sum\limits_{k = 0}^\infty {{x^k}} = \frac{1}{{1 - x}}\)
Using that we get \(\displaystyle F(-x) = \sum\limits_{k = 0}^\infty {{(-1)^k x^k}} = \frac{1}{{1 + x}}\)
Also \(\displaystyle F(x^2) = \sum\limits_{k = 0}^\infty {{x^{2k}}} = \frac{1}{{1 - x^2}}\)
Thus
\(\displaystyle \dfrac{1}{1+x^2}=\)\(\displaystyle \sum\limits_{k = 0}^\infty {{{( - 1)}^k}{x^{2k}}} \)...........(***)
Can you integrate both sides of (***)?

This may help or hurt.

 

[rir0302]

Thanks for your help! Another question though, what would the nth term be/mean? Is it different from the power series representation?

 

[pka]

Thanks for your help! Another question though, what would the nth term be/mean? Is it different from the power series representation?

There is always some confusion in terminology here. Part of the confusion is on the part of a sloppy lecturer (I confess to that).
Look at the series \(\displaystyle \sum\limits_{k = 0}^\infty {{{( - 1)}^k}{x^{2k}}} = \frac{1}{{1 + {x^2}}}\) .
There are basically two parts to that series: a sequence of terms \(\displaystyle u_n=(-1)^nx^{2n}\); and a sequence of partial sums \(\displaystyle S_n = \sum\limits_{k = 0}^{n } {{{( - 1)}^k}{x^{2k}}}\)
The \(\displaystyle n^{th}\) term is the nth term of sequence of terms.
If the sequence of partial sums converges that limit is the sum of the series. (it may be called the representation).