Revisiting the Convergence of Infinite Series in Calculus 2

GreenPrint
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Sense e^x=Ʃ[k=0,∞] x^k/k!
then
ln(e^x) = ln(Ʃ[k=0,∞] x^k/k!)
x = ln(Ʃ[k=0,∞] x^k/k!)

is this true?
 
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GreenPrint said:
Sense e^x=Ʃ[k=0,∞] x^k/k!
then
ln(e^x) = ln(Ʃ[k=0,∞] x^k/k!)
x = ln(Ʃ[k=0,∞] x^k/k!)

is this true?
Sure, but how useful it is, I don't know.

I sense that you don't understand the difference between sense and since.
 
Mark44 said:
Sure, but how useful it is, I don't know.

I sense that you don't understand the difference between sense and since.

I'm not sure that it is useful at all and is why I asked lol. Nope my English is god awful.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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