Thank you 12 Bonus Simplify the following formulas Sigma k1

Thank you!

12. (Bonus) Simplify the following formulas Sigma k=1 to n ar^k-1; Sigma k=1 to infinity ar^k-1;

Solution

a)

This in a geometric series with first term a1 = a, and common ratio R = r, with n terms.

Hence, as for a geometric series

Sum = a1 (1 - R^n) / (1 - R)

Then here,

Sum = a (1 - r^n) / (1 - r) [ANSWER]

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b)

Using the result in a, we see that as n--> infinity, r^n approaches 0, if -1<r<1. Hence,

Sum = a / (1 - r) [ANSWER]

or, divergent, if r is not -1<r<1.