- #1
CAF123
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Homework Statement
Consider the infinite series $$\frac{x}{e^x - 1} = A_o + A_1 x + \frac{A_2}{2!}x^2 + ... + \frac{A_n}{n!}x^n + ...$$ Determine that ##1 = A_o,\,\,\,\,\,0 = A_o/2! + A_1,\,\,\,\,\,0 = A_o/3! + A_1/2! + A_2/2!##.
Show that for ##n > 1##, one can write the relations as $$(A+1)^n - A^n,$$ where ##A^n \rightarrow A_n##.
2. Homework Equations
Taylor expansions and induction
The Attempt at a Solution
First part is fine. I thought about using induction on the second part since ##n## is confined to an integer. I understand ##A^n \rightarrow A_n## as meaning the powers of the series converge to the actual terms in the series, which is why I thought ##n## was an integer, labeling the terms in the series. However, I can't seem to make sense of the base case (n=2). It is $$(A+1)^2 - A^2 = 2A + 1$$ What is ##A##?
thanks