If $E$ is the spectral measure coresponding to a self-adjoint operator, show $\int1+2\sum_{i=1}^n\frac{n-i}nλ^idE(λ)f→\int\frac{1+λ}{1-λ}dE(λ)f$

Question

Let $(\Omega,\mathcal A,\mu)$ be a probability space, $\kappa$ be a contractive self-adjoint linear operator on $L^2(\mu)$ and $E:\mathbb R\to\mathfrak L(L^2(\mu))$ be the spectral family corresponding to $\kappa$. How can we show that $$\int 1+2\sum_{i=1}^n\frac{n-i}n\lambda^i\:{\rm d}E(\lambda)f\xrightarrow{n\to\infty}\int\frac{1+\lambda}{1-\lambda}\:{\rm d}E(\lambda)f\tag1$$ for all $f\in L^2(\mu)$? This looks like a simple application of Lebesgue's dominated convergence theorem, but since the sum in the integrand on the left-hand side tends to $\infty$, I'm confused.