106sir, how does the sequence diverge if cosθ is taken -1 ?
1+ \frac{cos\theta }{cos\theta } + \frac{cos2\theta }{cos^2\theta } + \frac{cos3\theta }{cos^3\theta } + ... upto \infty
for those whose latex is not working
1+cosθcosθ + cos2θcos2θ + cos3θcos3θ + ... upto ∞
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6 Answers
341Interesting question:
If cos θ = ±1,0 the series diverges.
Otherwise consider
1 + \frac{z}{\cos \theta} + \frac{z^2}{\cos ^2 \theta} +.... which I presume converges to \frac{1}{1 - \frac{z}{\cos \theta}}= \frac{i}{\tan \theta}
The given expression is Re(
1 + \frac{z}{\cos \theta} + \frac{z^2}{\cos ^2 \theta} +....) which is therefore 0
I am not sure about the conditions for the series to converge however
341I looked up the condition for convergence and I am beginning to have serious doubts about the problem.
A series like \sum \cos^n \theta \cos n \theta is surely convergent this way, but here it does seem to be divergent.
Is this question from a reliable source?
106oh sorry sir, but the answer was given as zero.. maybe they just calculated the answer without thinking about the conditions
341looks like that, because the sum to n terms would be sin nθ/cosn-1θ sin θ which doesnt converge