Can you construct a sequence of real nonzero numbers whose sum converges to 0?

[Bipolarity]

Does there exist a sequence of real nonzero numbers whose sum converges to 0?
I would think there isn't, but I'm interested in people's opinions and arguments.

For any nonzero m, a series of nonzero numbers whose sum converges to m can easily be constructed using the formula: $\sum ^{\infty}_{n=1}m(0.5)^{n}$

But that is for nonzero m, what if you wanted to construct a series whose sum converged to 0?

BiP

 

[Vargo]

1+-1+.5+-5+.25+-.25+.125+-.125+...

 

[Bipolarity]

1+-1+.5+-5+.25+-.25+.125+-.125+...

Can you find an explicit representation for that seqence (i.e. with sigma notation) ?

BiP

 

[Bipolarity]

Eureka! I believe I found it!

$$\sum^{\infty}_{n=1} (-1)^{n+1} (\frac{1}{2})^{ \frac{2n-3+(-1)^{n+1}}{4}}$$

I believe it converges to 0, but can anyone verify this?

BiP

 

[Edgardo]

If $\sum ^{\infty}_{n=1}m(0.5)^{n}$ converges to m, then shouldn't $m-\sum ^{\infty}_{n=1}m(0.5)^{n}$ converge to 0?

 

[Bipolarity]

If $\sum ^{\infty}_{n=1}m(0.5)^{n}$ converges to m, then shouldn't $m-\sum ^{\infty}_{n=1}m(0.5)^{n}$ converge to 0?

Yes, but $m-\sum ^{\infty}_{n=1}m(0.5)^{n}$ is not a series... unless you can express it as one with nonzero terms.

BiP

 

[lurflurf]

Who cares about expressing it as one with nonzero terms? A series is a series is a series.
This is simple
How about
$$\sum_{n=0}^\infty \frac{(\pi)^{2n+1}}{(2n+1)!} (-1)^n$$

 

[Bipolarity]

Who cares about expressing it as one with nonzero terms? A series is a series is a series.

The problem requires it.

BiP

 

[haruspex]

Yes, but $m-\sum ^{\infty}_{n=1}m(0.5)^{n}$ is not a series... unless you can express it as one with nonzero terms.

How about (1-λ)λⁿ-(1-μ)μⁿ for n >= 0, where λ is algebraic and μ is not?

 

[Millennial]

How about taking a sequence $(a_x)_x$ which satisfies $\displaystyle \lim_{x\to\infty}a_x = 0$ and then using the series $\displaystyle \sum_{x=0}^{\infty} (-1)^x b_x$, where the sequence $b_x$ is defined as $b_{2x} = b_{2x+1} = a_x$?

 

[lurflurf]

So you dislike the pi example and the usual example
$$\sum_{k=0}^\infty a_k b_x$$
where a_k is a sequence of positive numbers tending to zero and B_k is any sequence of -1 and 1 such that the series tends to zero.
What about any number of obvious examples such as
$$\sum_{k=0}^\infty (2k-1)\left(\frac{1}{3}\right)^k$$