# Summation (Again!!!)

What does $$1 + 2 + 3 + 4 + 5 + 6 + ...$$ equal?

We need to work with other identities first.

Let $$S_1 = 1 - 1 + 1 - 1 + ...$$;

and $$S_2 = 1 - 2 + 3 - 4 + ...$$.

First we solve $$S_1$$.

$$S_1 = 1 - 1 + 1 - ...$$

$$1 - S_1 = 1 - (1 - 1 + 1 - ...)$$

$$1 - S_1 = 1 - 1 + 1 - ...$$

$$1 - S_1 = S_1$$

$$S_1 = \frac {1}{2}$$

Now we solve $$S_2$$.

$$S_2 = 1 - 2 + 3 - 4 + ...$$

$$2S_2 = (1 - 2 + 3 - 4 + ...) + (1 - 2 + 3 - 4 + ...)$$

By shifting the addend one value to the right you get

$$2S_2 = 1 - 1 + 1 - 1 + ...$$

By using our previous value, we get

$$2S_2 =\frac {1}{2}$$

$$S_2 = \frac {1}{4}$$

Finally, we solve $$S$$

$$S = 1 + 2 + 3 + 4 + ...$$

$$S - S_2 = (1 + 2 + 3 + 4 + ...) - (1 - 2 + 3 - 4 + ...)$$

$$S - S_2 = 4 + 8 + 12 + ...$$

$$S - S_2 = 4 (1 + 2 + 3 + ...)$$

$$S - \frac {1}{4} = 4S$$

$$S = -\frac {1}{12}$$

Ta da!!!

Note by Sharky Kesa
4 years, 2 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

This is a famous result that $$\zeta (-1) = -\dfrac{1}{12}$$. (Saw this in Number Phile :P).

- 2 years, 4 months ago

Numberphile!!!!

- 3 years, 11 months ago

No, Wikipedia.

- 3 years, 11 months ago

oh!

I learnt it from that...

- 3 years, 11 months ago

Weird...I posted a discussion on these exact sums...

- 4 years, 2 months ago

when, because I want to read it.

- 4 years, 2 months ago

Here it is Interesting sums

- 4 years, 2 months ago

Wow. I found this when I was surfing the internet.

- 4 years, 2 months ago