Question

does the sequence n*tan(1/n) converge or diverge?
@OCW Scholar - Sin…

Answer 1

limit test, series diverges.

Answer 2

Guys, the question is does the limit as \( n \rightarrow \infty\) of n tan(1/n) exist.

Note first that

\[ n \tan(1/n) = \frac{\sin(1/n)}{1/n} \sec(1/n) \]

Now this limit does exist. The limit of the first term is 1 and the limit of the sec term is also 1.

Answer 3

You don't need to work so hard.

Since we're looking at n*tan(1/n) as n --> ∞ :

Assume x = 1/n That means we're looking for tan(x)/x as x approaches 0.

Using L'Hopitals rule, the solution is:\[\sec^2(x)/1 = 1\]

Answer 4

a) Both our proofs are just a couple of lines
b) One of uses artillery while the other just a sling shot. I think it's clear which is more elegant :-)

Answer 5

as james said the given question is \[\lim_{n \rightarrow \infty} n \tan (1/n)\]
read it as \[\lim_{n \rightarrow \infty} \tan(1/n)/(1/n)\]
lets make a substitution to solve the problems easier
make the substitution \[n \rightarrow 1/x \]
as \[n \rightarrow \infty \implies x \rightarrow 0\]
so the limit changes to \[\lim_{x \rightarrow 0} \tan x/x\] which is 1