Im unsure if this is just a stupid question because i have been independently studying this kind of math for about a week, but this has been bothering me lately as i have been exploring some definite integrals of the following:
$\int{e^{sin(x)}}dx $ and $\int{e^{cos(x)}} dx$
Im going to evaluate this function as an elementary example:
$$ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} $$ Therefore
$$ \frac{d}{dx} [e^x] = \frac{d}{dx} \sum_{n=0}^{\infty} \frac{x^n}{n!} $$
Which in turn is equal to $e^x$ which is a straight forward intuitive example, but what if you take another summation such as modified Bessel function of the first kind which takes this identity: $$ I_{v}(x) = \sum_{k=0}^{\infty} \frac{(\frac{x}{2})^{2k+v}}{k!\Gamma(k+v+1)} $$ Would i differentiate / integrate the summation with the same approach as with $e^x$, or is that method of differentiation a mathematical fallacy or a bad approach all together? If so would someone kindly point me to a good place to learn a good approach or answer my question i would appreciate it greatly.
Your method is good, and it is a very good question. Given a power series $\sum_{n=0}^\infty a_n (x - x_0)^n$ that converges for $|x - x_0| < R$ (where $R$ could be infinite), the sum of the series is an analytic function $f(x)$ on the interval $|x - x_0| < R$, and the derivative and indefinite integral on that interval can be obtained by differentiating or integrating term-by-term: $$ \eqalign{\dfrac{df}{dx} &= \sum_{n=0}^\infty n a_n (x - x_0)^{n-1}\cr \int f(x)\; dx &= C + \sum_{n=0}^\infty \dfrac{a_n}{n+1} (x - x_0)^{n+1}\cr}$$