General Cesaro summation with weight 
Assume that $a_n\to \ell $ is a convergent sequence of complex numbers and $\{\lambda_n\}$ is a sequence of positive real numbers such that $\sum\limits_{k=0}^{\infty}\lambda_k = \infty$

Then, show that, 
$$\lim_{n\to\infty} \frac{1}{\sum_\limits{k=0}^{n}\lambda_k} \sum_\limits{k=0}^{n}\lambda_k a_k=\ell =\lim_{n\to\infty} a_n$$ 
(Note that :  This is more general than the special case where, $\lambda_n= 1$)

 A: First you can assume that the $(a_n)$ are real, by considering 
the sequences
$(\operatorname{Re} a_n)_n$ and $(\operatorname{Im} a_n)_n$
separately.
And then it is an immediate application of the Stolz–Cesàro theorem to
$$
 A_n := \sum_{k=0}^n \lambda_n a_n \quad, \quad B_n := \sum_{k=0}^n \lambda_n
$$
since $(B_n)$ is strictly increasing and unbounded, and
$$
\frac{A_{n+1} - A_n}{B_{n+1} - B_n} = a_{n+1} \to l
$$
A: Let $\varepsilon >0$ and $N$such that $|a_k-l|\le \varepsilon $ for all $k>N$
Then, for $n>N$ we have,
\begin{split}\left|  \frac{\sum_\limits{k=0}^{n}\lambda_k a_k}{\sum_\limits{k=0}^{n}\lambda_k} -l\right|
&= &\left| \frac{\sum_\limits{k=0}^{n}\lambda_k (a_k - l)}{\sum_\limits{k=0}^{n}\lambda_k} \right|\\
&= &\left| \frac{\sum_\limits{k=0}^{N}\lambda_k (a_k - l)+\sum_\limits{k=N}^{n}\lambda_k (a_k - l)}{\sum_\limits{k=0}^{n}\lambda_k} \right|\\
&\le & \frac{M}{\sum_\limits{k=0}^{n}\lambda_k} + \frac{\sum_\limits{k=N}^{n}\lambda_k \underbrace{\left| a_k - l\right|}_{\le\varepsilon}}{\sum_\limits{k=0}^{n}\lambda_k} \\
&\le&
 \frac{M}{\sum_\limits{k=0}^{n}\lambda_k} + \varepsilon\to 0
  \end{split}
since $\sum_\limits{k=0}^{N}\lambda_k\to \infty$.
Where $M= \left|\sum_\limits{k=0}^{N}\lambda_k( a_k-l)\right|$
