Consider the series ∞Σk=-1 (5k+1) Write the series using sigma notation. and

Question

b. Find the sum S 100 of its first 100 terms, and then find an exact formula for the N-th partial sum. 

S100 =

c. Use the exact formula to determine if the series converges or diverges from the limit definition without using any tests of convergence/divergence. If it converges, find the value it converges to.

Answer 1

I see that you've asked for a sum with k ranging from -1 to infinity. You almost never see an initial k-value of -1 and instead often see an initial value of k=1 so I'm going to assume that's what you meant. If you meant k=-1, feel free to correct me in a response.

You've already begun to write the sum in sigma notation but it can more formally as lim_{b->∞} b∑k=1 (5k + 1).

Let's look at an exact formula for partial sums:

See that b∑k=1 (5k + 1) is the same as 5 * (b∑k=1 k) + b∑k=1 1. You may have learned that b∑k=1 k is equal to (b(b+1))/2 and that b∑k=1 1 is equal to b. Therefore, your whole formula is (5b(b+1))/2 + b. Let's find S100 which is where b=100. The sum is 5*100*101/2 + 100 = 25350.

Looking at this formula, we can see that the sum must diverge as the formula only gets bigger and bigger as b increases, with no end. Formally, we can say that lim_{b->∞} (5b(b+1))/2 + b = ∞ so the sum diverges.