How to approach expanding f(x) = int[0, x] [ln(1 + t^2)] dt in a power series

[Sophdof1]

I having great difficult with fundamental theorem of calculus so I am now even more confused when they ask me to expand that function f(x) in a power series

 

[Steven G]

You need to integrate the power series.

Do you know the power series for ln(x) or maybe ln (1+x)? How can this help?

 

[Sophdof1]

This is what I have so far but idk how to use it

 

[Steven G]

I'll repeat what I said a bit more slowly.
1st step: find the power series for ln(1+t²). naturally you get an infinite series.
2nd step: What ever infinite series you got in step 1 you should now integrate.
If you need further help please reply back.

 

[Sophdof1]

Hi - only issue now is.. they ask me to obtain the value of the function at x=1... idk how that is possible since it is an infinite sum. Unless they are wanting me to find the infinite sum when x=1.


Thanks a lot for your help

 

[Steven G]

You really need to state the entire problem at the start. Maybe not in this case, but in theory the technique to solve this problem might have changed knowing that x=1.

You work looks good although I briefly looked at it.

One comment I would make is that you expanded much too early. You could have simply integrated (-1)^n-1(x²ⁿ.

The only problem I see is that you did not finish. What is the value of this infinite sum?

 

[Sophdof1]

You really need to state the entire problem at the start. Maybe not in this case, but in theory the technique to solve this problem might have changed knowing that x=1.

You work looks good although I briefly looked at it.

One comment I would make is that you expanded much too early. You could have simply integrated (-1)^n-1(x²ⁿ.

The only problem I see is that you did not finish. What is the value of this infinite sum?

I am not sure how I am supposed to find the infinite sum - I am rather unsure

 

[Steven G]

I don't think it will be easy. I think all your attempts will fail, but you never know. Most importantly you have to try! I would first try partial fractions and see what cancels out.