Find a power series representation for f, and graph several partial sums sn(x) on the same screen for f(x)=tan^-1(2x). What happens as n increases? (Yes it gets more accurate) As per homework...

Find a power series representation for f, and graph several partial sums sn(x) on the same screen for f(x)=tan^-1(2x). What happens as n increases? (Yes it gets more accurate)

As per homework solutions #24 section 11.9 8th edition Probably somewhere in the 7th edition too:

Yes that is the derivative above^

What did they do to get here^? (Definitions, words, series tests)

Again what did they do to get here^? If it is a mistake that is okay just fix it. (Definitions, words, series tests)

Again what did they do to get here^? (Definitions, words, series tests)

Again what did they do to get here^? (Definitions, words, series tests)

I DO see^ that simplication that is clear and makes sense.

"This series converges when 

What did they do to get here^? (Definitions, words, series tests) I think it is the ratio test but usually below like you did for the alternating series test you should say by the ratio test the series converges.

"So the interval of convergence is [-1/2, 1/2] because the alternating series test"

Is it [-1/2, 1/2] or (-1/2, 1/2)? I do think the alternating series^ test and some other tests like the ratio test, limit test, direct comparision test, show end point inclusion or exclusion. But again it needs a more clear explanation.

Graphing the rest well I'm fine with that graphing makes sense but none of the stuff before that does.

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

It is known that the derivative of arctan(x) is `1/(1+x^2).` This function may be expressed as `sum_(k=0)^oo (-1)^k x^(2k),` because this sum is the sum of a geometric series with the common ratio `-x^2.` The radius of convergence is 1 by the Ratio test, for example. Inside (-1,1) the series may be integrated, so `arctan(x)=sum_(k=0)^oo (-1)^k x^(2k+1)/(2k+1)` for -1<x<1.

The radius is also 1. At `x=+-1` the series is also convergent by the alternating series test: signs are changing and the magnitude 1/(2k+1) is monotone decreasing.

It is also the question whether arctan(x) is equal to the sum at x=+-1. The answer is yes: arctan is continuous and the sum on [-1,1] is also continuous, because the series converges at the endpoints (there is the second Abel theorem and its corollary).

For arctan(2x) the same is true by substituting 2x instead of x. Then x in [-1/2,1/2].

For graphing I suggest desmos.com

Borys Shumyatskiy eNotes educator| Certified Educator

The graph:

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wein7145 | Student

Again only part of it is showing up.

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I don't think my images showed up here:

Find a power series representation for f, and graph several partial sums sn(x) on the same screen for f(x)=tan^-1(2x). What happens as n increases? (Yes it gets more accurate)

As per homework solutions #24 section 11.9 8th edition Probably somewhere in the 7th edition too:

`tan^-1(2x)=int(2)/(1+4x^2)dx`

Yes that is the derivative above^

`intsum_(n=0)^oo2(-4x^2)^ndx`

What did they do to get here^? (Definitions, words, series tests)

Again what did they do to get here^? If it is a mistake that is okay they just need to fix it. (Definitions, words, series tests)

`intsum_(n=0)^oo(-1)^n2^(2n+1)x^(2n)dx`  

Again what did they do to get here^? (Definitions, words, series tests)

Again what did they do to get here^? (Definitions, words, series tests)

I DO see^ that simplication that is clear and makes sense.

"This series converges when `|_(-4x^2)_|<1rArr|_x_|<1/2`

What did they do to get here^? (Definitions, words, series tests) I think it is the ratio test but usually below like you did for the alternating series test you should say by the ratio test the series converges.

"So the interval of convergence is [-1/2, 1/2] because the alternating series test"

Is it [-1/2, 1/2] or (-1/2, 1/2)? I do think the alternating series^ test and some other tests like the ratio test, limit test, direct comparision test, show end point inclusion or exclusion. But again it needs a more clear explanation.

Graphing the rest well I'm fine with that graphing makes sense but none of the stuff before that does.

Your help is making things more clear now.