Sequences and series (1)

Algebra Level 5

Find the sum of the terms of the Geometric Progression, a+ar+ar2+ar3+a+ar+ar^2 + ar^3 + \cdots, where aa is the value of xx for which the function 7+2xln255x152x7+2x \ln 25 -5^{x-1} - 5^{2-x} has a greatest value and rr is equal to limx00xt2dtx2tan(π+x) \displaystyle\lim_{x\rightarrow 0} \displaystyle\int_{0}^{x} \frac{t^2 dt}{x^2 \tan (π+x)}


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