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

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

#### sID='l2-HINT2';writePM(sID)startA('s'+sID)Hint

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

#### sID='l2-HINT3';writePM(sID)startA('s'+sID)Hint

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

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

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

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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.

#### sID='l2-HINT7';writePM(sID)startA('s'+sID)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.

#### sID='l2-HINT8';writePM(sID)startA('s'+sID)Hint

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

#### sID='l2-HINT9';writePM(sID)startA('s'+sID)Hint

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