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Use the deterministic weights $a_{n,k}=\sqrt{k}/n$. Their squared sum and maximum areTo apply the quadratic expansion of a cumulant-generating function rigorously, write $\log M_X(u)=u^2/2+r(u)$, with $r(u)=o(u^2)$. Fix $t\ne 0$ and letAll arguments $t\sqrt{k}/n$ lie in this shrinking interval. ConsequentlyTaking logs of the moment-generating function and using independence of random variables now givesThe case $t=0$ is immediate. Since $e^{t^2/4}$ is the moment-generating function of a normal distribution of variance $1/2$, the convergence theorem for moment-generating functions givesThis is a weighted central limit theorem with negligible maximum weight. Its limit variance also agrees with the exact value $\mathrm{Var}(T_n)=(n+1)/(2n)$ from variance additivity for independent random variables.

The given expected value and variance imply $M_X(0)=1$, $M_X^{\prime }(0)=0$ and $M_X^{\prime \prime }(0)=1$. The Taylor expansion therefore givesSince $\log (1+u)=u+o(u)$ as $u\to 0$, it follows thatThis is the quadratic expansion of a cumulant-generating function for a centered variable of variance one.

Let $X_k$ be independent and identically distributed random variables with expected value zero, variance one, and a moment-generating function finite in a neighborhood of zero. Each row of deterministic real weights $a_{n,k}$ is finite. If ${\max }_k|a_{n,k}|\to 0$ and ${\sum }_k{a}_{n,k}^2\to v<\infty$, then the weighted sums converge in distribution to a normal distribution with mean zero and variance $v$; $v=0$ denotes the constant zero.

By the quadratic expansion of a cumulant-generating function, write $K(u)=u^2/2+r(u)$ with $r(u)=o(u^2)$. For a fixed real $t$, put ${\eta }_n={\sup }_{0<|u|\le |t|{\max }_k|a_{n,k}|}|r(u)|/u^2$, with the supremum taken as zero if its range is empty. Then ${\eta }_n\to 0$ andThus the log moment-generating function of the sum tends to $v{t}^2/2$. The convergence theorem for moment-generating functions proves the conclusion. The supremum is the step that makes the remainder estimate uniform across the row; a separate little-o assertion for each term would not justify summing an increasing number of terms.