Home > Calculus > Limit Comparison Test for Convergence of an Infinite Series

Find the limit of the sequence an=2^n/(2n-1)?

Answer 1

#lim_(n->oo)2^n/(2n-1)=oo#

So, we want #lim_(n->oo)2^n/(2n-1)=oo/oo#

This is an indeterminate form and does not tell us anything of value.

We could solve by applying l'Hospital's Rule, but a sequence is not differentiable, so we need to define a function in terms of #x.#

Let #f(x)=2^x/(2x-1)#

Then, differentiating numerator and denominator, we obtain

#lim_(x->oo)2^x/(2x-1)=lim_(x->oo)(2^xln(2))/2=oo/2=oo#

Thus,

#lim_(n->oo)2^n/(2n-1)=oo#

The sequence diverges to #oo#

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Answer 2

Charles Arocha

The limit of the sequence (a_n = \frac{2^n}{2n - 1}) as (n) approaches infinity is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.