Question

How do you determine whether the infinite series ${{a}_{n}}={{e}^{\dfrac{1}{n}}}$ converges or diverges?

Answer 1

For problems regarding convergence or divergence we first of all need to understand the idea behind the sums. Solving the problem will be very easy if we get the core concept. We have many tests for testing the convergence or divergence of a series or an infinite series. Some of these tests include d’Alembert test, Raabe’s Test, Cauchy’s root test and many more. For any of these we need to be fluent in some of the basics as well as advanced topics of limit, continuity and differentiability. Here in this problem for the test of convergence we need to check what happens to the value of the expression when the value of ‘n’ tends to infinity. If this results in a definite value then we can clearly understand that this expression is convergent. However if it results in an indefinite value then we need to apply another test for convergence. If all of them fail then the series is divergent.

Complete step-by-step solution:
Now we start off with the solution to the given problem by trying to find the value of the expression when the value of ‘n’ tends to infinity. So in mathematical form we write it as,

& \displaystyle \lim_{n \to \infty }{{a}_{n}} \\\ & =\displaystyle \lim_{n \to \infty }{{e}^{\dfrac{1}{n}}} \\\ \end{aligned}$$ Now in this equation we put the value of ‘n’ as infinity and then we get, $$\begin{aligned} & =\displaystyle \lim_{n \to \infty }{{e}^{\dfrac{1}{\infty }}} \\\ & =\displaystyle \lim_{n \to \infty }{{e}^{0}} \\\ & =1 \\\ \end{aligned}$$ Thus, the value of the limit yields $$1$$ , which is a definite value and thus from this we can clearly say that this given expression converges and we do not need to test for any other further tests. **Note:** These types of problems require a thorough knowledge of differential calculus and its various applications. We first of all need to find out the limiting value of the problem when the series approaches infinity. If this results in a definite value then we say it’s convergent, else we proceed further for the other tests. Once all the possible tests fail we say that the series is divergent and it doesn’t yield a definite value.