Let $s_{N}=\sum_{i=1}^{N}a_{i}$ a partial sum of real numbers. Let $L=\lim_{i\to +\infty}a_{i+1}/a_{i}$. Suppose $L$ finite and nonzero. The well known ratio criterion describes the convergence of the sequence $\{s_{N}\}$. Is there any way to approximate $s_{N}$ as a function of $L$ and $a_{N}$ for $N>>0$? Something like $s_{N}\thicksim f(L,a_{N})$ with some fixed function $f$? In case yes, are there also explicit behaviours of $s_{N}$ in terms of the root $\lim_{N\to +\infty}a_{N}^{1/N}$?
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closed as offtopic by Alexandre Eremenko, RP_, Franz Lemmermeyer, Myshkin, Sebastian Goette Apr 17 '17 at 12:57
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1$\begingroup$ If you could, that would imply something about the first few terms. However, $a_1$ is arbitrary. How do you want this to behave if you change just $a_1$ to $10$, or $10^{10}$, or $10^{10^{10}}$? Gerhard "Let Series Remain Wild, Free" Paseman, 2017.04.16. $\endgroup$ – Gerhard Paseman Apr 17 '17 at 3:54

$\begingroup$ Lets take for example $\sum_{i=1}^{N}q^{i}$. In this case $s_{N}=\frac{qq^{N+1}}{1q}$ (let us take $q\neq 1$). Here $L=q$ so we see an expression $f(a_{N},L)$. What about for example $\sum_{i=1}^{N}\frac{q^{i}}{i}$? $\endgroup$ – Hair80 Apr 17 '17 at 8:33

$\begingroup$ Note that you are considering very special examples here. Just replacing $q^i$ by $aq^i$ in the first example already spoils your $f(a_N,L)$ for $0<q<1$. $\endgroup$ – Sebastian Goette Apr 17 '17 at 12:57