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There will be no difference in the answer whether you solve it using L.Hospital's rule or not. But yeah, there obviously exists a way to evaluate it without using L.Hospital's rule, as Pranshu did. And notice that the answer is still the same.
@Sandeep Bhardwaj
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We could just use L'Hopital's rule but sometimes we can search about another answer ...I was looking for an approach that does not use this rule and I am sorry I didn't mention , but your solution still elegant :))
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
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[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"\(...\)or\[...\]to ensure proper formatting.2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
Without L'Hôpital's rule
Rationalize the denominator by multiplying the expression with x+1x+1. Distribute the n and rewrite the expression as
x→1limx−1(x−1)+(x2−1)+(x3−1)+⋯+(xn−1)⋅(x+1)
=x→1lim(x−1x−1+x−1x2−1+x−1x3−1+⋯+x−1xn−1)⋅(x+1)
=x→1lim((1)+(1+x)+(1+x+x2)+(1+x+x2+x3)+⋯+(1+x+x2+x3+⋯+xn−1))⋅(x+1)
Now solve by substitution:
=(1+2+3+⋯+n)⋅(1+1)=n(n+1)
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Amazing solution! :)
n2+n???
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Did you use L.Hospital's Rule ?
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Yes. Can u prove the L.hospital rule?
n(n+1)
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Did you use L.Hospital's Rule ?
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Yes. However I cannot post my solution. My Latex skills s*cks. Sorry.
n(n+1) May be thats the answer. If it is correct please notify.
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Did you use L.Hospital's Rule ?
n(n+1)
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Did you use L.Hospital's Rule ?
n(n+1)
Since this is an indeterminate form of the 00 form, we can simply use the L.Hospital's Rule to evaluate the given limit, which is as you follow:
\[\begin{array}{} & \lim_{x\to1} \dfrac{x+x^2+x^3+\ldots +x^n-n}{\sqrt x-1} \\ & = \lim_{x \to 1} \dfrac{1+2x+3x^2+\ldots +nx^{n-1}}{\dfrac{1}{2 \sqrt x}} \\ & = n(n+1) \end{array} \]
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There is an answer without using L.Hospital's Rule ,....can u find it ?
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There will be no difference in the answer whether you solve it using L.Hospital's rule or not. But yeah, there obviously exists a way to evaluate it without using L.Hospital's rule, as Pranshu did. And notice that the answer is still the same.
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We could just use L'Hopital's rule but sometimes we can search about another answer ...I was looking for an approach that does not use this rule and I am sorry I didn't mention , but your solution still elegant :))
n(n+1)