Exercises

1. Using the Integral Test, determine whether the following infinite series converge or diverge:

   1. 

   2. 

   3. 

   4. 

2. The items in this problem refer to the following series:

   

   1. Determine the appropriate function for this series and then calculate the appropriate integral. Explain why this shows that the series converges.
   2. Plot the partial sums for . Why does this plot show that ? Is it always true that for this series?
   3. How many terms of the series must be added together to guarantee that the remainder is no more than ?

3. Consider the harmonic series

   

   1. Use the integral test to show that this series diverges.
   2. Compute and/or plot partial sums, , of this series for various values of n. How large must n be for the partial sum to be greater than 20? Greater than 50? Do you think that if you just computed partial sums and didn't use the integral test you would think the series converged?