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Unity is confusing!

$\huge 1^{1+2+3+4+5+6+7\cdots}$

Considering the fact that some diverging sums can also approach a certain limit. What is the sum of real value(s) of the expression above?

Details:

• If no real value(s) are obtained give your answer as $$Not$$ $$defined$$.

• If the limit approaches $$\infty$$ then enter your answer as $$\infty$$ as well.

This problem is original. Upon pondering over the answer to this problem I wasn't able to come up with a legit explanation as to why my method was/wasn't correct. Please enter your answer with an appropriate explanation.

Note by Tapas Mazumdar
2 weeks ago

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We are essentially calculating $$\displaystyle \lim_{n\to\infty} 1^{n(n+1)/2} = 1^\infty = 1$$. · 1 week, 5 days ago

Also for any real number $$n$$ (say) we can say that:

$$\displaystyle \lim_{x\to\infty}{\large\sqrt[x]{n}} = 1$$

$$\because$$ The above can be written as $$\displaystyle \lim_{x\to\infty}{\large n^{\frac{1}{x}}}$$.

$$\therefore$$ As $$x\to\infty$$, $$\dfrac{1}{x}\to 0$$, hence limit of the function reaches $$1$$.

So, if $$\sqrt[\infty]{n}=1$$ (as in most cases, the limit and the value mean quite the same thing), can't we say that $$1^{\infty} = n$$, which is kind of a paradox. And the only plausible explanation to this is that $$1^{\infty}$$ is Not defined.

But if we follow $$\zeta\left(-1\right)=\dfrac{-1}{12}$$, then this turns out to be $$\large1^{\frac{-1}{12}}$$ which I'm confused about because the $$12$$th root of $$1$$ is both $$1$$ and $$-1$$ (or is it just 1?) and $$10$$ complex roots.

So, what can be the answer? · 1 week, 5 days ago

Read up indetermediate forms, you did not obey those rules, so your logic is incorrect.

Plus, if you want to invoke 1+2+ 3 + ... = -1/12, then you should make it clear that you're using riemann zeta regularization from the start. Otherwise, by convention, 1 +2 + 3 + ... = infinity · 1 week, 2 days ago

But the string theory tells us that $$\displaystyle \sum_{n=1}^{\infty}{n}~ \left(\text{or}\right)~ \zeta\left(-1\right) = \dfrac{-1}{12}$$. Wouldn't that concept be counted right here? · 1 week, 5 days ago

Think about how zeta(-1) = -1/12 was derived in the first place. Did it apply Abel sum? Read up sums of divergent series. · 1 week, 2 days ago