Hey guys, I have a few problems that if possible, I would love to have someone check my solutions to them.

**Problem 1.** Determine whether the improper integral

converges, or diverges. In the case of convergence, give its value.

**My Solution** ... Converges to ln2

Nop: and thus the integral diverges. Of course, you can't write ... **Problem 2.** Determine whether the sequence with the given general term converges, or diverges. In the case of convergence, give its value.

**a.)** **My Solution** , Using L'Hopital's Rule.

... Converges to

Nop. You can't use DIRECTLY L'Hospital with a discrete variable since L'H implies the use of derivative which use limits which need a continuous variable. Of course, you can use L'H with and then use that the limit stays the same no matter how , and thus this is so if you choose to go to the limit along the naturals. Another way, perhaps more natural , to divide both numerator and denominator by the highest power of n and use arithmetic of limits: Tonio **b.)** **My Solution** No Limit, Diverges

**c.)** **My Solution** ...

Using L'Hopital's Rule.

...

... Converges to e

**Problem 3.** Calculate the sum of the given convergent geometric series.

**a.)** ...

**My Solution** ,

**b.) ** **My Solution** ,

**Problem 4.** Write the first three terms of the sequence of partial sums of the series

.

**My Solution** ,

,

Thank you in advance for any help!