Question

Which of the following series converges?

I. \(\mathop \sum \limits_{n = 1}^\infty \frac{{3 + \cos n}}{{{e^n}}}\)

II. \(\mathop \sum \limits_{n = 1}^\infty \cos \left( {\frac{1}{n}} \right)\)

1. I only
2. II only
3. I and II both
4. neither I no II

Answer 1

Correct Answer - Option 3 : I and II both

Concept:

Consider the infinite series ∑un = u1 + u2 + u3 + … ∞

Convergence test:

If un tends to a finite or unique limit as n → ∞, the series is said to be convergent

If un tends to ± ∞ as n → ∞, the series is said to be divergent

If un does not tend to a unique limit as n → ∞, the series is said to be oscillatory or non-convergent

Calculation:

Given

(I) \(\mathop \sum \limits_{n = 1}^\infty \frac{{3 + \cos n}}{{{e^n}}}\)

Check for n = ∞

\(\mathop {\lim }\limits_{n \to \infty } \frac{{3 + \cos n}}{{{e^n}}}\)

The value of cos(∞) is either +1 or -1 so take is as ‘K’

\( = \mathop {\lim }\limits_{n \to \infty } \frac{K}{{{e^\infty }}} = \frac{K}{\infty } = 0\)

Hence series 1 convergent.

(II) \(\mathop \sum \limits_{n = 1}^\infty \cos \left( {\frac{1}{n}} \right)\)

Check for n = ∞

\( = \mathop {\lim }\limits_{n \to \infty } \mathop \sum \limits_{n = 1}^\infty \cos \left( {\frac{1}{n}} \right) = \cos \left( {\frac{1}{\infty }} \right)\)

Cos (0) = 1 

Hence series (2) is convergent