Series
I used this for Problem 10
MATH147-Midterm_practice_problems1.pdf > page=2
Since it hasn’t been covered in my 147 class or is in the Notes. I will have to first proof this theorem from scratch before using it.

Proof

Suppose that we have a series and either or . i.e. is an alternating series. Then converges if (Not iff)

2. is decreasing

Note: Conditions 1 and 2 imply that

Note: For this proof we are assuming , with small tweaks we can readjust it for the case . We can also just multiple the entire series by .

Because

Because


Because

Therefore we know that is an increasing sequence. We can write the general term as

Since each of the terms in the parenthesis are positive and so is we can say that is bounded by . Since is bounded and monotonic (decreasing). We can say that it converges by the Monotone Convergence Theorem (MCT)

Since converges, let

Therefore since both and converge. We can say that converges.