1. Find the power series expansion of \displaystyle{\frac{\ln(1+t)}{1-t}} as far as the term in t^4.

Hint

2. Let \displaystyle{\sin(2x)\cos(x) = \sum_{n=0}^\infty a_n x^{2n+1}}. Find a_n explicitly.

Hint

3. Let \displaystyle{\ln(1-x+x^2) = \sum_{n=1}^\infty a_n x^n}. Find a_n explicitly.

Hint

4. If \displaystyle{\frac{\ln(1+t)}{1-t}} is expanded in powers of t, what will the radius of convergence be?

Hint

5. If e^x \cos(x) is expanded in powers of x, what will the radius of convergence be?

Hint

6. Find explicitly the sum of the series \displaystyle{\sum_{n=1}^\infty n^2 x^n}.

Hint

7. Let \displaystyle{\left(\frac{1+t}{1-t}\right)^\frac{1}{2} = \sum_{n=0}^\infty a_n t^n}. Find a_n explicitly.

Hint

8. For the series \displaystyle{\sum_{n=0}^\infty \frac{(2n+1)!}{2^n (n!)^2} x^n}, find the sum and the interval of convergence.

Hint

9. Let f(x) = x^{-1}\sin(x^2) if x \neq 0 and f(0)=0. Find f^{(9)}(0).

Hint

10. Find the expansion of e^{(\cos(x))^2} in powers of x as far as the term in x^8.

Hint