Sequences and Series – Infinite Zeros in Infinite Series

Question

The problem:

Given that

$$\frac{\pi}{4} = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + \frac{1}{9} + \ldots $$

Prove

$$\frac{\pi}{3} = 1 + \frac{1}{5} – \frac{1}{7} – \frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \ldots$$

My solution:

We know

$$
\begin{align}
\frac{\pi}{4} & = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + \frac{1}{9} -\frac{1}{11} + \frac{1}{13} – \frac{1}{15} + \ldots \\
\\
\frac{\pi}{12} & = \frac{1}{3} – \frac{1}{9} + \frac{1}{15} – \frac{1}{21} + \frac{1}{27} -\frac{1}{33} + \frac{1}{39} – \frac{1}{45} + \ldots\\
\\
& = 0 + \frac{1}{3} + 0 + 0 – \frac{1}{9} + 0 + 0 + \frac{1}{15} + 0 + 0 – \frac{1}{21}
\end{align}
$$

now add them together:

$$
\begin{align}
\frac{\pi}{4} + \frac{\pi}{12} & = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} + \frac{1}{9} – \frac{1}{11} + \frac{1}{13} – \frac{1}{15} + \ldots \\
\\
& + 0 + \frac{1}{3} + 0 + 0 – \frac{1}{9} + 0 + 0 + \frac{1}{15} + \ldots \\
\end{align}
$$

and we will get:

$$
\begin{align}
\frac{\pi}{3} & = 1 + 0 + \frac{1}{5} – \frac{1}{7} + 0 -\frac{1}{11} + \frac{1}{13} + 0 + \ldots \\
& = 1 + \frac{1}{5} – \frac{1}{7} -\frac{1}{11} + \frac{1}{13} + \ldots
\end{align}
$$

My questions:

• I inserted/removed infinite zeros into/from the series, is that OK?

• My solution relies on the fact that $\Sigma a_n + \Sigma b_n = \Sigma (a_n + b_n)$ and $k \Sigma a_n = \Sigma k a_n$. Is this always true for convergent infinite series? If so, why is it? (yeah I know this is a stupid question, but since I'm adding infinite terms up, I'd better pay some attention.)

• Bouns question: Can I arbitrarily (arbitrariness isn't infinity, you know) insert/remove zeros into/from a convergent infinite series, without changing its convergence value?

Answer 1

1. Yes, that is ok.
2. Yes it is. The partial sum of $\sum_k (a_k+b_k)$ is the partial sum of $\sum_ka_k$ plus the partial sum of $\sum_k b_k.$ The result follows from the sum property for limits.
3. Yes. Adding zeros will only delay the inevitable convergence of the sequence of partial sums. Where you insert zeros, the sequence of partial sums will hold flat. For the $N$ you find in the proof of convergence of the original, simply replace with $N$ plus the number of zeros you inserted before the N-th and you'll have the same value that will be within $\epsilon$ of the number the sum converges to.