MHB Investigate Convergence of tanx Series: Find Common Ratio & Sum to Infinity

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The discussion focuses on investigating the convergence of the sequence tan(x), tan(2x), tan(3x), ..., tan(nx) for x in the interval (-90, 90 degrees). The series can be expressed as a geometric series, which converges when the absolute value of tan(x) is less than 1, specifically for -π/4 < x < π/4. The sum to infinity of the series is given by the formula 1/(1 - tan(x)). There is confusion regarding the evaluation of this sum at x = 45 degrees, where tan(45) equals 1, leading to division by zero. The conclusion emphasizes leaving the sum expressed as 1/(1 - tan(x)) for values of x that do not cause divergence.
Alexeia
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Hi,

Please help me with this question: Investigate the convergence of the sequence tanx;tan2x;tan3x;...;tannx for xE(-90;90 degrees). Steps to follow: Find common ratio. Draw the graph. For which values will x converge. Determine sum to infinity.

I did try to solve, but file type too big to upload my answers.

Please help..

Thanks
 
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Alexeia said:
Hi,

Please help me with this question: Investigate the convergence of the sequence tanx;tan2x;tan3x;...;tannx for xE(-90;90 degrees). Steps to follow: Find common ratio. Draw the graph. For which values will x converge. Determine sum to infinity.

I did try to solve, but file type too big to upload my answers.

Please help..

Thanks

Setting $\tan x = \xi$ the series to be analized is $\displaystyle \sum_{n = 0}^{\infty} \xi^{n}$, which is 'geometrical' and converges for $|\xi|< 1 \implies -\frac{\pi}{4} < x < \frac{\pi}{4}$... in case of convergence is $\displaystyle \sum_{n=0}^{\infty} \tan^{n} x = \frac{1}{1 - \tan x}$...

Kind regards

$\chi$ $\sigma$

P.S. The formula for geometric sums is in...

Geometric Series -- from Wolfram MathWorld
 
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Thank you,

The last part, how do you derive that the sum to infinity is 1 \div(1 - tanx)? Is it according to the Sum to infinity formula? To find the answer do I use 1 \div(1 - tan(45))? 1\div1 - tan(45) , Cant divide by 0.. ? Or do I just leave it as 1\div(1 - tanx)
 
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