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Can anyone find the error in this? Or is this just another divergent series?

  • How is the first step valid? $\Pi_{i=1}^n i^3$ diverges. – Zach Effman Apr 12 '15 at 15:39
  • @ZachEffman I chose to thought of everything as a partial sum/product, and then the limit as they all (uniformly) go to infinity. e.g. the partial term $s_n$ is where the sum and both products go from $1$ to $n$. It makes sense then – MT_ Apr 12 '15 at 15:44
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    The last step reads: $$\frac{\infty}{\infty}=1\ \text{which means}\ \infty=1.$$ After such a piece of creativity, anything can ensue. What is the source of this? – Did Apr 12 '15 at 16:11

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In the last $=$ on the first line, you move the $\displaystyle \sum_{k=1}^{\infty}$ term to the right, yet you keep it in the quotient by accident. After that step you should get

$$\frac{\prod_{i=1}^{\infty} i^3}{\prod_{k=1}^{\infty} k^3}=1.$$

Also, it's bad practice to use the same variable in two different sum/product notations. It makes it very confusing what "k" is being incremented how.

MT_
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$$a_1b_1 + a_2b_2 + a_3b_3 + \dots \neq (a_1 + a_2 + a_3 + \dots)(b_1+b_2+b_3+\dots)$$

In other words, $\sum a_ib_i \neq \sum a_i \sum b_j$

You wrote $$\sum[(\prod [i^3]) \frac{1}{k^3}] = (\sum\prod i^3 )\cdot\sum \frac{1}{k^3}$$ which is invalid.

JMoravitz
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  • the sum increments $k$, so I would chock it up more to an accident/confusion of notation (since the sum has no effect on the product variable) than a misunderstanding of concept – MT_ Apr 12 '15 at 16:04