Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals.
QUESTION. What is the value of this limit? It seems to exist. $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\lambda_1+\lambda_2x)^n-x^n}{1-x}\,dx.$$
Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals.
QUESTION. What is the value of this limit? It seems to exist. $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\lambda_1+\lambda_2x)^n-x^n}{1-x}\,dx.$$
The limit is $-\ln(\lambda_{2})$.
\begin{align*}
\int_{0}^{1}\frac{(\lambda_{1}+\lambda_{2}x)^{n}-x^{n}}{1-x}dx
&= \int_{0}^{1}\frac{(\lambda_{1}+\lambda_{2}x)^{n}-1}{1-x}dx + \int_{0}^{1}\frac{1-x^{n}}{1-x}dx \\
&= \int_{0}^{1}\frac{(1-\lambda_{2}t)^{n}-1}{t}dt + \int_{0}^{1}\sum_{k=0}^{n-1} x^{k-1}dx \\
&=\int_{0}^{1}-\lambda_{2}\sum_{k=0}^{n-1}(1-\lambda_{2}t)^{k}dt + \sum_{k=0}^{n-1}\frac{1}{k+1} \\
&=\sum_{k=0}^{n-1}\frac{(1-\lambda_{2})^{k+1}-1}{k+1} + \sum_{k=0}^{n-1}\frac{1}{k+1} = \sum_{k=0}^{n-1}\frac{(1-\lambda_{2})^{k+1}}{k+1} \\
&\to -\ln(\lambda_{2}).
\end{align*}