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Let $0 \not\neq p(x) $ be a nondecreasing real valued function on $[0, \infty)$ such that $p(0)=0$. For a  real valued function $f(x)$ which is continuous on $[0, \infty)$, we define\begin{equation}\label{1}s(x)=\int_0^x f(t) dt,\end{equation}and$$\sigma_{p}(x)=\frac{1}{p(x)}\int_0^x p'(t)s(t) dt,$$where $p'(t)$ is derivative of $p(t).$ If $\displaystyle{\lim_{x \rightarrow \infty} \sigma_p (x)=s}$ then the improper integral $\int_0^{\infty} f(t) dt $ is said to be weighted mean summable to a finite number $s$. It is known that if the  limit $\displaystyle{\lim_{x \rightarrow \infty}s(x)=s}$ exists, then  $\displaystyle{\lim_{x \rightarrow \infty} \sigma_p (x)=s}$ also exists. However, the converse is not always true.Adding some suitable conditions to  weighted mean summability of $s(x)$ which are called Tauberianconditions may imply convergence of the integral (\ref{1}).
In this work, we give some Tauberian theorems to retrieve convergence of $s(x)$ out of weighted mean summability of $s(x)$ with some Tauberian conditions.

Tauberian theorem, Tauberian condition, Weighted mean method, slowly oscillatingsequence, slowly decreasing sequence, one-sided condition.