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Maths-

General

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Question

The value of $sum from n equals 1 to straight infinity of fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction$ is

$\frac{2}{\sin 2 θ} - \frac{1}{θ}$
$\frac{2}{\sin 2 θ} + \frac{1}{θ}$
$\frac{1}{\sin 2 θ} - \frac{1}{θ}$
$\frac{1}{\sin θ} - \frac{1}{θ}$

hint Hint:

In this question, we have to find the value of $sum from n equals 1 to straight infinity of fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction$ . For this first we will simplify the trigonometric function. Then as we have to find the sum of infinity terms, so, we can use the limit and so solving the sima and the limit we can get the required value.

The correct answer is: $fraction numerator 2 over denominator sin space 2 theta end fraction minus 1 over theta$

$G i v e n comma space sum from n equals 1 to straight infinity of fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction fraction numerator tan begin display style space end style open parentheses theta over 2 to the power of n close parentheses over denominator 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator sin begin display style theta over 2 to the power of n end style over denominator 2 to the power of n minus 1 end exponent cos begin display style theta over 2 to the power of n end style cos space begin display style theta over 2 to the power of n minus 1 end exponent end style end fraction m u l t i p l y i n g space n u m e r a t o r space a n d space d e n o m i n t a o r space b y space 2 sin theta over 2 to the power of n equals fraction numerator 2 sin squared theta over 2 to the power of n over denominator 2 to the power of n minus 1 end exponent space 2 sin theta over 2 to the power of n cos theta over 2 to the power of n cos space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 1 minus cos theta over 2 to the power of n minus 1 end exponent over denominator 2 to the power of n minus 1 end exponent space sin theta over 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 1 minus cos theta over 2 to the power of n minus 1 end exponent over denominator 2 to the power of n minus 2 end exponent space sin theta over 2 to the power of n minus 2 end exponent end fraction equals 1 over 2 to the power of n minus 2 end exponent open parentheses fraction numerator 1 over denominator sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator cos space theta over 2 to the power of n minus 1 end exponent over denominator 2 sin space theta over 2 to the power of n minus 1 end exponent cos space theta over 2 to the power of n minus 1 end exponent end fraction close parentheses equals fraction numerator 1 over denominator 2 to the power of n minus 2 end exponent sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction N o w comma space sum from n equals 1 to straight infinity of fraction numerator 1 over denominator 2 to the power of n minus 2 end exponent sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction equals limit as n rightwards arrow infinity of open parentheses sum from r equals 1 to straight infinity of fraction numerator 1 over denominator 2 to the power of r minus 2 end exponent sin space theta over 2 to the power of r minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of r minus 1 end exponent sin space theta over 2 to the power of r minus 1 end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of open parentheses fraction numerator 1 over denominator 2 to the power of negative 1 end exponent sin space 2 theta end fraction minus fraction numerator 1 over denominator sin space theta end fraction plus fraction numerator 1 over denominator sin space theta end fraction minus fraction numerator 1 over denominator 2 sin space begin display style bevelled theta over 2 end style end fraction plus....... fraction numerator 1 over denominator 2 to the power of n minus 2 end exponent sin space theta over 2 to the power of n minus 2 end exponent end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of open parentheses fraction numerator 2 over denominator sin space 2 theta end fraction minus fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction close parentheses equals limit as n rightwards arrow infinity of fraction numerator 2 over denominator sin space 2 theta end fraction minus limit as n rightwards arrow infinity of fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent sin space theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 2 over denominator sin space 2 theta end fraction minus limit as n rightwards arrow infinity of fraction numerator 1 over denominator 2 to the power of n minus 1 end exponent begin display style fraction numerator sin space theta over 2 to the power of n minus 1 end exponent over denominator theta over 2 to the power of n minus 1 end exponent end fraction end style cross times theta over 2 to the power of n minus 1 end exponent end fraction equals fraction numerator 2 over denominator sin space 2 theta end fraction minus limit as n rightwards arrow infinity of 1 over theta equals fraction numerator 2 over denominator sin space 2 theta end fraction minus 1 over theta$

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