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For each continuous function $f \in C[0,1]$ define the continuous function $T(f)(x)$ by

$$T(f)(x)=x^2 + \int_{0}^{x}tf(t)dt$$

for each $x\in C[0,1]$. I'm trying to find the power series to represent function f(x) and I am not sure I/m on the right track. I worked the sequence and am getting a pattern that looks like

$$x^2 + \frac{x^4}{4} + \frac{x^6}{4\cdot6} + \frac{x^8}{4\cdot6\cdot8} + ...$$

I've come up with

$$f(x) = x^2 + \sum\limits_{i=1}^{n-2} \frac{x^{2i+2}}{4\cdot6\cdot8\cdot\cdot\cdot(2i+2)}$$

Am I on the right track? Where have I gone wrong?

Maggy Wood
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1 Answers1

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Nothing is gone wrong. The limit function is: $$ f(x) = x^2 + \sum_{i=1}^{+\infty}\frac{x^{2i+2}}{2^i(i+1)!} = 2\left(e^{x^2/2}-1\right).$$

Jack D'Aurizio
  • 353,855