Just two separate sums have been merged through. Anyways, bravo Sir!!
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+ar^{2}.......âˆž 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)}

UP 0 DOWN 0 0 2
2 Answers
for x part.. put 5^{x} = 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]
cheers!!