Given a sequence $\{a_i\}$ such that $a_1=1$ and
$a_{n+1}=a_n+\sqrt{1+a_n^2} \;\;\;\;\; \forall{n \in {\mathbb{N}}}$Calculate $$\lim_{n\to \infty}\frac{a_n}{2^n}$$
I had the answer with me. And it was $\frac{2}{\pi}$
So it came obvious to me that I had to use a trigonometric substitution. So I tried $\tan{\theta}$ and it became messy. The sequence was easy to solve on substituting $\cot{\theta}$.
I used $$\cot{\theta}+\csc{\theta}=\cot{\frac{\theta}{2}}$$
where $$\theta=\frac{\pi}{4}$$
Giving $$a_n=\cot{\frac{\frac{\pi}{4}}{2^{n-1}}}$$
and it was easy to solve ahead.
The problem is I am preparing for a competitive exam and I won't have the answer key there. It wasn't at all intuitive to me that I had to go for a trigonometric substitution? Can I approach this question in any other manner?
Also, How do I get an idea where trigonometric substitutions will make the calculations easier?
Let me know, If I am missing some details. I will add them.
Thank you.
Best Answer
This seems like an opinion based question. Personally speaking you never know what would be intuitive when encounter problems in the test room. This kind of competition, i assumed, is aiming at testing your familiarity of basic methods and your ability of observance, i.e. your ability to spot the breakthrough point of designed problems. Then, of great probability, you would repeat the “trial and error” process until you find a feasible way to continue.
As for this question, if I have not encountered questions of this type ever, then the trigonometric substitution would be something come out from nowhere [at least for myself].
And for such a short time I do not know any alternative way to do this. Personally speaking I might let $b_n = a_n 2^{-n}$ and let $n \to \infty$ but this would yield nothing since we would obtain an equality.
As I experienced, $\sqrt {1\pm t^2}$ would make me try trig substitution. Same thing when I encountered some equation taking a similar form to, say formula of double angles and triple angles. Anyway the more you prepared, the more thoughts come up when dealing the test.