The geometric series N-1 s = La n S n=0

The geometric series

will be used quite frequently in the next chapters so let us look at some of its properties:

(a) Suppose α = 1 what is S equal to?

(b) Suppose α ≠ 1 show that

Verify that (1 − α)S = (1 − α^N). Why do you need the constraint that α ≠ 1? Would this sum exist if α > 1? Explain.

(c) Suppose now that N = ∞, under what conditions will Sexist? if it does, what would S be equal to? Explain.

(d) Suppose again that N = ∞ in the definition of S. The derivative of S with respect to α is

obtain a rational expression to find S₁.

Transcribed Image Text:

N-1 s = La" n S n=0