GATE CSE 1993 | Question: 02.2

Question

The radius of convergence of the power series$$\sum_{}^{\infty} \frac{(3m)!}{(m!)^3}x^{3m}$$ is:   _____________

Comments

can someone please answer this question

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What is written below the cursor in the image ?

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'm!' is written

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@Satbir,@Sherrinford, starting value of 'm' is not given in the question.

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lets assume m starts from zero

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then answer will be $1/3$.. check here :-

https://math.stackexchange.com/questions/3298415/radius-of-convergence-for-a-power-series

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Thank you for Answer, are these of questions in GATE 2020 syllabus?

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Radius of Convergence topic comes under power series which is a part of calculus. Since, power series is not mentioned in the syllabus, So, I guess chances are very less that these type of questions will be asked in gate but for other exams like ISI, TIFR etc, chances are very high :P

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But , it came in GATE . So, better to do. right??

With stirling formula it came $3.$ Then why radius of converge inverse the value??

and u initialized  $m=0$, but it can be $m=1$ too. right??

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But , it came in GATE . So, better to do. right??

I think syllabus of calculus was different that time. Please check some questions of 1993 below, It involved Fourier series, double integral etc.

With stirling formula it came 3. Then why radius of converge inverse the value?? 

please check it : https://en.wikipedia.org/wiki/Radius_of_convergence#Theoretical_radius

and u initialized  m=0, but it can be m=1 too. right??

yes.

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its part of numerical methods which is no more in GATE CS syllabus.

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@ankitgupta.1729

@Satbir

 Say, for an equation $a_{n}$, if the equation converges, then  $\lim_{n\rightarrow \infty}a_{n}=0.$ But here it comes $3,$ which means here function  diverges. So, to make it convergent, we inverses it. 

right??

Please check some questions of 1993 below, It involved Fourier series, double integral etc.

limit, convergence-divergence, double integral is in syllabus. These thing are not out of syllabus. But fourier series not in syllabus now.