I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$
$f_1:\Re \rightarrow \Re$ is defined as follows
$$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$
One sees that the sequence has to be:
$$f_{n+1} (x) = 1 + \int_0^x f_n (t) dt$$
Now I want to compute the value of these functions. One gets:
For $n = 0$
$$f_{1} (x) = 1 + \int_0^x f_0 (t) dt$$
(the provided one).
For $n = 1$
$$f_{2} (x) = 1 + \int_0^x f_1 (t) dt$$
For $n = 2$
$$f_{3} (x) = 1 + \int_0^x f_2 (t) dt$$
$$...$$
Basically what I am doing here is iterating and what I have finally got is:
Knowing that $f_n : \Re \rightarrow \Re: x \mapsto 1$
$$f_{1} (x) = 1 + x$$
$$f_{2} (x) = 1 + x + \frac{x^2}{2}$$
$$f_{2} (x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$
$$...$$
Thus the sequence $(f_n)_n$ does not seem to converge while it should...
What's wrong with my reasoning?
I am also curious on how could I prove the convergence of a sequence which includes an integral on it. I have been studying the convergence of a sequence theorem and understood it. I have applied it to straightforward examples but never to a sequence with an integral on it. How could I do so?
EDIT:
So we get that
$$f_{n+1} (x) = \sum_{n}^{N} x^{n}/n!$$