Use this list of basic taylor series to find the taylor series for f(x) = 2x 1+x2 based at 0. give your answer using summation notation and give the largest open interval on which the series converges. (if you need to enter ∞ , use the ∞ button in calcpad or type "infinity" in all lower-case.) the taylor series for f(x)= 2x 1+x2 is: ∞ k=0 the taylor series converges to f(x) for |x|< . use the taylor series you just found for f(x)= 2x 1+x2 to find the taylor series for ln(1+x2) based at 0. give your answer using summation notation and give the largest open interval on which the series converges. the taylor series for ln(1+x2) is: ∞ k=0 the taylor series converges to ln(1+x2) for |x|< .

Following are the calculation to the Taylor Series:

Geometric series:

which converges to .

Remembering that:

 Taking  

Using the Taylor series:

In the given scenario we will converge the .  Now, realize:

Integrating the series for  :

Since integrating a number has no effect on its radius of converge, this series similarly converges for .

Learn more about Basic Taylor Series:

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Recall that for , we have

Then for , or , we have

Multiply this series by to get the Taylor series for :

Notice that

so to find the Taylor series for , we integrate the Taylor series for :

Since , it follows that and

which converges for as well.

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step-by-step explanation:

through just a quick inspection, one can see that the graph of this cubing function starts in quadrant iii and ends in quadrant i.   when x = 0, y = 3, so the y-intercept is (0, 3).

if graphs are part of the problem statement, share them.   .