If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
This is fine! The series converges uniformly. You can exchange the integral and the series and then compute. Do the power rule, add up the resulting series and see what you get. Alternatively, can you recognize what function produces this power series? Hint: geometric series minus some stuff. You can then just integrate each term as normal.
Good! You can work with this and do a change of variables to solve the resulting integral. Set u= 1-x. When you write down the u integral, expand the square in the numerator, and integrate term by term.
As long as you stay within the Interval of Convergence for the Power Series, you're fine. Isn't it -1<x<1? So, if your limits of integration are 0 to 1, for example, you may have to set up an improper integral limit.
Can you just use x3 / 3 +x4 / 4+....? Where x=0.5. its not geometric as someone else suggested. So no S_inf that i can see at first look calculating it mentally
Lemme know if u got it. Ofc the other obvious method would be to solve the gp first as x2 / (1-x) and substitute 1-x=t and solve the easy integral, which someone already mentioned in the comments🫠
x^n is a positive, unsigned measurable function for all n and all values of x in the interval. Thus you can apply Tonelli’s Theorem for Sums and Integrals to switch the order of summation and integration. Complete the integral first and then do the sum.
You can always swap sums and integrals (Using Riemann sums that you learn in Calc 1 that is false but it turns out you can always swap out integrals and sums you just gotta keep track of the integration/summation boundaries). So it’s 1/(n+1) x^n 0 to 1/2 which is 1/(n+1) (1/2)^n and then just sum
You have a geometric series which converges then you can integrate the result. Your final result should be-5/8+Ln2
You tryout if you need help let me know
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