Define the sequence $I_n$ as: $I_n=\int_0^1x^n\exp\bigg(\frac{x}{5}\bigg)dx.$
Let $I_0'$ be an erroneous starting value and $f_n(I_0)=I_n$ as well as $f_n(I_0')=I'_n$ be two functions that are to be evaluated. Find the absolute error in $$|f_n(I'_0)-|=|I_n'-I_n|. $$
I know that $I_0=5(\exp(1/5)-1)$ and also that $I_{n+1}=5\exp(1/5)-5(n+1)I_n$ holds true, but I'm not sure how to proceed when I don't know what the error is. I thought of this: Letting $x'=x+\epsilon$
$$|I_n'-I_n|= \bigg| \int_0^1 (x+\epsilon)\exp(\frac{x+\epsilon}{5})-x\exp(\frac{x}{5})dx\bigg| $$ While I could make some algebraic manipulations I couldn't go any further. I'd appreciate help
Using the recursion formula you obtained, you can write down the successive errors: $$ \begin{align*} |I_1'-I_1| = & 5|I_0'-I_0|\\ |I_2'-I_2| = & 5\times 2 \times |I_1'-I_1| = 5^2 \times 2 \times |I_0'-I_0|\\ |I_3'-I_3| = & 5 \times 3 \times |I_2'-I_2| = 5^3\times 3 \times 2 \times |I_0'-I_0|\\ \vdots & \end{align*} $$
and, ignoring round off errors, you can "guess" that $$|I_n'-I_n| = 5^n n! \,\,|I_0'-I_0|.$$
As an example, imagine that we set $I_0' = 1.10701$, which satisfies $|I_0'-I_0| \leq 0.38 \times 10^{-5}$. The error in $I_8'$, computed using the exact solution, is $|I_8'-I_8| \approx 59705.1$, which is exactly what this formula yields.