AP Calculus BC - 10.2: Summing an Infinite Series

How do you test for convergence in an infinite series?

You create an entirely new sequence filled with each Nth partial sum of the infinite series. You then find the general term to represent each Nth partial sum. You then take the limit of that general term. If it exists, then the series converges, if it doesn’t, then the series diverges.

how can I rewrite sum(a_n) + sum(b_n)?

As sum(a_n + b_n)

how can I rewrite sum(a_n) - sum(b_n)

As sum(a_n - b_n)

how can I rewrite sum(c * a_n)

As c * sum(a_n)

How do you know if an infinite geometric series diverges or converges?

If |r| < 1, then it converges

If |r| => 1, then it diverges

How do you find the sum of a geometric series? What MUST be true in order for you to find the sum?

You can only do this if it converges. Then the sum is (cr^M)/(1-r), where M is the first n value for the series. E.g. if we defined the series to start at term 2, then M would be 2

What is the divergence test?

When testing a series for convergence, if it doesn’t converge to 0, then the series diverges