TITLE: Fake proof of the limit of a series
QUESTION [5 upvotes]: Now, I know this to be correct:
$$\begin{align*}
\lim_{n \rightarrow\infty} \left(\frac 1{n^2}+\frac 2{n^2}+\ldots+\frac n{n^2}\right)&=\lim_{n \rightarrow\infty} \left[\frac 1{n^2} \left(\frac n2\right)(1+n)\right]\\
&=\lim_{n \rightarrow\infty} \frac {1+n}{2n}\\
&=\frac 12\;.
\end{align*}$$
But what is wrong with the following reasoning?
$$\begin{align*}
\lim_{n \rightarrow\infty} \left(\frac 1{n^2}+\frac 2{n^2}+\ldots+\frac n{n^2}\right)&=\lim_{n \rightarrow\infty} \frac 1{n^2} + \displaystyle \lim_{n \rightarrow\infty} \frac 2{n^2} +...+ \displaystyle \lim_{n \rightarrow\infty} \frac n{n^2}\\\\
&=0+0+\ldots+0\\\\
&=0\;?
\end{align*}$$

REPLY [5 votes]: Look at $$\frac1{n^2}+\frac2{n^2}+\ldots+\frac{n}{n^2}$$ for the first few values of $n$:
$$\begin{array}{c}
\frac11\\
\frac14&+&\frac24\\
\frac19&+&\frac29&+&\frac39\\
\frac1{16}&+&\frac2{16}&+&\frac3{16}&+&\frac4{16}\\
\vdots&&\vdots&&\vdots&&\vdots&&\ddots
\end{array}$$
It’s true that each column is converging to $0$, but the number of columns is increasing at the same time, and you don’t know until you work it out which effect is going to ‘win’. In this case neither wins: they almost exactly balance out, with an overall limit of $1/2$.
It’s somewhat analogous to $0\cdot\infty$ indeterminate forms, limits of the form $\lim\limits_{x\to\infty}f(x)g(x)$ where $\lim\limits_{x\to\infty}f(x)=0$ and $\lim\limits_{x\to\infty}g(x)=\infty$: here $f$ can win, making the limit $0$, or $g$ can win, making it $\infty$, or they can roughly balance out, making it some positive real number.