In the text "Function Theory of One Complex Variable" Third Edition by Robert E.Greene and Steven G.Krantz. I'm having trouble proving the following case of the root test in $(1.)$


$(1.)$

$\text{Lemma 3.2.6 (The root test)}:$ The radius of convergence of the power series $\sum_{} a_{k}(z-P)^{k}$ is



$$ \frac{1}{\lim \sup_{k \rightarrow \infty}|a_{k}|^{\frac{1}{k}} }$$

$(a)$ if $\lim \sup_{k \rightarrow \infty}|a_{k}|^{\frac{1}{k}} > 0$, or

$(b)$
if $\lim \sup_{k \rightarrow \infty}|a_{k}|^{\frac{1}{k}} = 0$

In the text "Real and Complex Integration Problems" I'm having trouble verifying my approach to proving following conjecture in $(1.) - (2.).$


$(0.)$


$(0.1)$

$$K_{n}(\theta) = \sum(1 - \frac{|j|}{n+1})$$


$(1.)$

Show that for any continuous $2 \pi$ periodic function $f$ on $\mathbb{R}$ one has:
$$ K_{n} \star f(\theta) := \frac{1}{ 2 \pi} \int_{}^{} K_{n}(\theta - a)f(\theta)d \theta = \sum(1 - \frac{|j|}{n+1})(\frac{1}{2 \pi} \int_{}^{}e^{-ija}f(\alpha) d \alpha)e^{ij\theta}$$


$(2.)$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, …

Can anyone check if the initial result is correct here, I tired a new way of dealing with definite integrals via change of variables for the integral defined in $(1.)$



$(1.)$
$$\frac{1}{2 \pi}\int_{\pi}^{-\pi} \frac{\sin((N+\frac{1}{2})\theta)}{\sin \frac{\theta}{2}}$$

Lemma:

To handle calculations of $(1.)$ more gently one considers the indefinite case as follows in $(1.5)$

$$\frac{1}{2 \pi}\int\frac{\sin((N+\frac{1}{2})\theta)}{(u)} $$

$$u = sin(\frac{\theta}{2})$$
$$du = \frac{d}{d \theta}\sin({\frac{\theta}{2}})$$