Math with Mrs. Sarah

(Finney, AP Calculus BC)

Lots of ratio test here
Finding the radius of convergence
Finding the interval of convergence
Finding the sum of the infinite series
Finding the sum of the telescoping series

1. Find the values of x for which the equation is an identity:
   −1/(x+4) = 1 + (x+5) + (x+5)² + (x+5)³ + …

Find the values of x for which the equation is an identity:
-1/(x+4) = 1 + (x+5) + (x+5)^2 + (x+5)^3 + …

2. Use a comparison test to show that ∑ (n=0 to ∞) x⁴ⁿ / (2n! + 1) converges for all x.

Use a comparison test to show that sum(x^(4n) / (2n! + 1), n=0 to inf) converges for all x.

3. Show that the series ∑ 2(sinx)ⁿ / (n! + 3) converges absolutely.

Show that the series summation(2(sin(x))^n / (n! + 3), n=0 to infinity) converges absolutely.

4. Find the radius of convergence of ∑ aₙ, where aₙ= (3x−2)ⁿ / n

Find the radius of convergence of summation((3x-2)^n / n, n=1 to infinity)

5. Find R of ∑ aₙ, where aₙ= nxⁿ / (n+2)

Find R of summation(nx^n / (n+2), n=0 to infinity)

6. Find R of ∑ aₙ, where aₙ= x^(2n+1) / n!

Find R of summation(x^(2n+1) / n!, n=0 to infinity)

7. Find R of ∑ nxⁿ / (4ⁿ(n²+1))

Find R of summation(nx^n / (4^n(n^2+1)), n=0 to infinity)

8. Find R of ∑ √n · xⁿ / 3ⁿ

Find R of summation(sqrt(n) * x^n / 3^n, n=0 to infinity)

9. Find R of ∑ (4x−5)^(2n+1) / n^(3/2)

Find R of summation((4x-5)^(2n+1) / n^(3/2), n=1 to infinity)

10. Find R of ∑ (x − √2)^(2n+1) / 2ⁿ

Find R of summation((x – sqrt(2))^(2n+1) / 2^n, n=0 to infinity)

11. Find the interval of convergence and the sum as a function of x.
∑ (x+1)²ⁿ / 9ⁿ

summation((x+1)^(2n) / 9^n, n=0 to infinity)

12. Find the interval of convergence and the sum as a function of x
∑ (lnx)ⁿ

summation((ln(x))^n, n=0 to infinity)

13. Find the interval of convergence and the sum as a function of x
∑ (sinx / 2)ⁿ

summation((sin(x)/2)^n, n=0 to infinity)

14. Determine the convergence or divergence of the series ∑ (2ⁿ / (n+1)

summation(2^n / (n+1), n=1 to infinity)

15. Determine the convergence or divergence of the series ∑-1/8ⁿ

summation(-1/8^n, n=1 to infinity)

16. Determine the convergence or divergence of the series ∑ n sin(1/n)

summation(nsin(1/n), n=1 to infinity)

17. Determine the convergence or divergence of the series ∑ n! / nⁿ

summation(n!/n^n, n=1 to infinity)

18. Find the sum of the telescoping series ∑ 4 / ((4n−3)(4n+1))

summation(4/((4n-3)(4n+1)), n=1 to infinity)

19. Find the sum of the telescoping series ∑ 40n / ((2n−1)² (2n+1)²)

summation(40n/((2n-1)^2 * (2n+1)^2), n=1 to infinity)