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x+x+x+x+x+...... Up to finite x times= x^2

Differentiating both sides we get 1+1+1+1+1......=2x What is wrong in this?? Attempt: I'm thinking that the above statement written is only for natural numbers..

Vidit Kulshreshtha
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1 Answers1

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I've seen this question before, but apparently not on math.SE. I'm a bit surprised it's not a duplicate; it's quite a common thing to come up with.

The trick is in the sneaky "…" you included. How does one differentiate "…"?

Consider instead the very similar question:

$1+1+\dots+1 = x$. So if we differentiate both sides, we should get $0+0+\dots+0$ on the LHS, and $1$ on the RHS. What have I done wrong?

The answer is that you've tried to differentiate a sum whose length varies with $x$. To do this properly, you're going to have to use the chain rule. Linearity of differentiation only works when the length of the sum is fixed in advance.

Patrick Stevens
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  • OK let's say I have 100 times x being summed up... Now is my "..." clear?? – Vidit Kulshreshtha Sep 02 '16 at 09:56
  • @ViditKulshreshtha Yes, but now your "100" is fixed in advance, so the differentiation will work because differentiation is linear. Basically you're using "…" to hide the fact that you're misusing the $+$ symbol: you're using $+$ to define an operation ("add $x$ times") which is similar to, but not identical to, the usual $+$ ("add once"). Then you express surprise that the rules of differentiation of $+$ don't hold with your new operation. – Patrick Stevens Sep 02 '16 at 10:29