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Help in limits maybe?

Here's a problem i am not being able to solve:

\(prove\) \(that\)

\[\large{lim_{x\to \frac{\pi}{2}}}\LARGE{(1^{\frac{1}{cos^{2}x}}+2^{\frac{1}{cos^{2}x}}+...........+n^{\frac{1}{cos^{2}x}})^{cos^{2}x}=n}\]

Please help

Note by Aritra Jana
3 years, 1 month ago

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To prove :

\[\displaystyle \large{lim_{x\to \frac{\pi}{2}}}\LARGE{(1^{\frac{1}{cos^{2}x}}+2^{\frac{1}{cos^{2}x}}+...........+n^{\frac{1}{cos^{2}x}})^{cos^{2}x}=n}\]

Let \(y=\frac{1}{cos^2x}\) . As \(x\to \frac{\pi}{2}, y\to \infty\).

Therefore , \[limit=\displaystyle Lim_{y\to \infty} (1^y+2^y+3^y+....+n^y)^{\frac{1}{y}} \quad \quad \quad \infty^0 \ form \] \[\quad \quad \quad = (n^y)^{\frac{1}{y}} \left[ \left(\frac{1}{n} \right)^y+ \left(\frac{2}{n} \right)^y+ \left(\frac{3}{n} \right)^y+......+ \left(\frac{n-1}{n} \right)^y+1 \right]^{\frac{1}{y}}\] \[ \quad \quad \quad = n.\left[ \left(\frac{1}{n} \right)^y+ \left(\frac{2}{n} \right)^y+ \left(\frac{3}{n} \right)^y+......+ \left(\frac{n-1}{n} \right)^y+1 \right]^{\frac{1}{y}}\] \[\quad \quad \quad =n . (1)^0 =\boxed{n}\]

Enjoy @Aritra Jana !

Sandeep Bhardwaj - 3 years, 1 month ago

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ohhh.. silly me! i forgot to consider that each of \(\frac{i}{n}\) for \(i≤n-1\) is less than \(1\)!!

making \(\lim_{y\to\infty}(\frac{i}{n})^{y}=0\)

anyways. thanks a lot for replying :D

Aritra Jana - 3 years, 1 month ago

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