I've discovered through Wolfram Alpha that
∑∞t=1e−bt=1eb−1
What are the steps of derivation here? According to infinite summation of power series:
∑∞t=1pt=11−p,
I expected the solution to be
∑∞t=1(e−b)t=11−e−b.
What am I getting wrong?
In extension, how do I derive
∑∞t=1e−b(t−1) ?
Answer
Your second formula isn't quite right: if |p|<1, then
∞∑t=1pt=p1−p
Using this with p=e−b yields
∞∑t=1e−bt=e−b1−e−b=1eb−1
as claimed.
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