In graph x there is an input of an data set 1, which computes to a line a, which is approximated with line b.
How can I proof that line b is an approximation of line a and goes on infinite?
EDIT here the my try to proof it
Given the function $$ \delta (x) = \frac{\gamma(x+1)}{\gamma(x)} $$
To show that $$ \lim x\rightarrow\infty \sum_{x=1}^{\infty} \delta (x) = \infty $$ Let k be the limit of $$\delta (x) $$ $$ \lim x\rightarrow k \sum_{x=1}^{\infty} \frac{\gamma(x+1)}{\gamma(x)}=k $$ Be $$x=k$$ $$c +\frac{\gamma(k+1)}{\gamma(k)}=k \rightarrow \gamma(k)= \frac{\gamma(k+1)}{k}+\frac{c\gamma(k+1)}{k}=\frac{2c\gamma(k+1)}{k} $$ Be $$x = k + 1$$ $$\gamma(k+1)=\frac{2c\gamma(k+1)\gamma(k+2)}{k}=k \rightarrow k= \frac{2c\gamma(k+1)\gamma(k+2)}{\gamma (k+1)} $$ $$ c +\frac{\gamma(k+1)}{\gamma(k)} \neq 2c\gamma(k+2) $$
qed
Is this proof correct?