For Subho

Not exactly Olympiad stuff, infact I don't know if it is. I just needed a section to post it. Others may try and solve it if they please -:

Find the sum of terms of the GP : a+ar+ar2.......∞ where a is the value of x for which the function :

f(x) = 7 + 2xlog(25) - 5^{x - 1} - 5^{2 - x}

has the greatest value.

r is given by the following limit -:

lim_{x \to 0} \int_{0}^{x}\frac{t^2dt}{x^2tan(\pi + x)}

2 Answers

Subhomoy Bakshi ·

for x part.. put 5x = t
then the t at whch fuunc becomes max gives us the corresponding x...
then party...

take a while's rest.. come back to r...

It actually is \lim_{x\rightarrow 0}\frac{\int_{0}^{x}{t^2dt}}{x^2tan(\pi +x)}

Next.. we need to realise the above limit is of 0/0 form.. apply L'Hospital..

for differentiating numerator we need to use the basic principle of calculus,
If\; F(x)=\int_{0}^{x}{f(t)dt}; \\then,\; F'(x)=f(x)

phir wapas party shuru [3]


Vivek @ Born this Way ·

Just two separate sums have been merged through. Anyways, bravo Sir!!

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