Linked Questions

12
votes
5answers
4k views

Prove $ 1 + 2 + 4 + 8 + \dots = -1$ [duplicate]

Possible Duplicate: Infinity = -1 paradox I was told by a friend that $1 + 2 + 4 + 8 + \dots$ equaled negative one. When I asked for an explanation, he said: Do I have to? Okay so, Let $x ...
1
vote
5answers
169 views

Sum of $1+2+4+8+…$ [duplicate]

I was solving a recurrence problem which had a sequence such as $y = (1+2+4+8+...)\sqrt n$, and I wanted to find what $x = 1+2+4+8+...$ was. So consider $x = 1+2+4+8+...$ as an infinite series. $$x-1 ...
3
votes
0answers
234 views

Proof that $ -1 = \infty $ . [duplicate]

Possible Duplicate: Infinity = -1 paradox $1+2+4+8+16+\ldots = \infty$ $LHS=1(1+2+4+8+16+\dots)$ $LHS=(2-1)(1+2+4+8+16+\ldots)$ $LHS=(2+4+8+16+32+\ldots)-(1+2+4+8+16+\ldots)$ $LHS=2+4+8+16+32+...
-2
votes
1answer
93 views

Can anyone Find the error below [duplicate]

If: $$S=1+2+4+8+....+2^n +...$$ So we get $$2S=2+4+8+...$$ $$2S+1=1+2+4+8+16...$$ $$2S+1=S$$ $$2S-S=-1$$ $$S=-1$$ Is there error, and if there's, why? I want athletic explanation.
2
votes
0answers
179 views

Is $\sum \limits_{n=0}^{\infty}2^n=-1$? [duplicate]

Possible Duplicate: Infinity = -1 paradox MinutePhysics has what initially looks like a divergent series summing to -1. The youtube comments are... lacking in clarity. The argument MinutePhysics ...
2
votes
0answers
106 views

Paradox of Infinity? [duplicate]

If a series such as '$a$' below adds to infinity: $a = 1 + 2 + 4 + 8 + 16 + \cdots\to \infty$ Multiplying '$a$' by $2$ yields: $2a = 2 + 4 + 8 + 16 + \cdots\to \infty$ However when I subtract ...
1
vote
0answers
73 views

A weird infinity problem. [duplicate]

A weird infinity problem. I saw this on youtube but could not understand it: Let us add 1 + 2 + 4 + 8 + 16 + ... up to infinity x=(1+2+4+8+...) = 1(1+2+4+8+...) = (2-1)(1+2+4+8+...) = (2+4+8+16+...)-(...
1
vote
0answers
40 views

Is “shift & add” technique valid for infinite sequences? [duplicate]

The "shift & add" technique is an excellent trick to get standard formulas for summation of various finite sequences. e.g. (Newbie here, couldn't get the aligning proper, so stuck to special case ...
380
votes
15answers
48k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
20
votes
5answers
2k views

Prove that the infinite sum $\sum_{n=1}^{\infty} \frac{F_{n}}{ 10 ^ n }$ converges to a rational number

How do you prove that the following infinite sum \begin{align} &0.1 \\+\;&0.01 \\+\;&0.002 \\+\;&0.0003 \\+\;&0.00005 \\+\;&0.000008 \\+\;&0.0000013 \\ \;&\quad\...
7
votes
3answers
1k views

Understanding power series and their representation of functions

Below is my current level of understanding about Power Series (my understanding could be completely wrong, in which case please correct me), and I want to know if it is correct. I feel that Power ...
1
vote
3answers
264 views

Why do we pick $\frac{1}{1-x}$ for $f(x)=1+x+x^2+x^3+…+x^n$, where $n$ tends to infinity?

When we consider the function $f(x)=1+x+x^2+x^3+...+x^n$ where $n$ tends to infinity, we can rewrite this as $$f(x)=1+x(1+x+x^2+x^3+...)=1+x(f(x))\qquad (1)$$ After some algebraic manipulations, we ...
10
votes
1answer
358 views

Is there an extension of the integers where the “sum of natural numbers” is rigorous?

There's the well-known claim that $$\sum_{n=1}^{\infty} n = -\frac{1}{12} \tag 1$$ Of course in this form, using the usual interpretation of the infinite sum as limit of finite sums, it's wrong, as ...
-3
votes
2answers
135 views

Why it is so? Please explain please help me [duplicate]

Why $ \infty - \infty \neq 0 ?$ Please explain
0
votes
0answers
125 views

Ramanujan's claim [duplicate]

I was reading about the mathematician Ramanujan, and he claimed that 1+2+3+4+5+...=-1/12, and that this is related to the Riemann Zeta Function. Can someone explain the relationship? (By the way, I ...