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

[Alexeia]

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

 

[chisigma]

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

 

[Alexeia]

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)