Question

Which series is convergent? Check all that apply.

Answer 1

The convergent series are;

`\sum \limits _(n = 1)^\infty \left( (1)/(5) \right) ^n`

(And)

`\sum \limits _(n = 1)^\infty \left( (1)/(10) \right) ^n`

Explanation:

A series in mathematics is the sum of a sequence of numbers to infinity

A convergent series is a series that sums to a limit

From the given options, we have;

First option

`\sum \limits _(n = 1)^\infty (2\cdot n)/(n + 1)`

As 'n' increases, 2·n becomes more larger than n + 1, and the series diverges

Second option

`\sum \limits _(n = 1)^\infty (n^2 - 1)/(n - 2)`

As 'n' increases, n² - 1, becomes more larger than n - 2, and the series diverges

Third option

`\sum \limits _(n = 1)^\infty \left( (1)/(5) \right) ^n`

As 'n' increases,
`\left( (1)/(5) \right) ^n`, becomes more smaller and tend to '0', therefore, the series converges

Fourth option

`\sum \limits _(n = 1)^\infty \left( (1)/(10) \right) ^n`

As 'n' increases,
`3 *\left( (1)/(10) \right) ^n`, becomes more smaller and tend to '0', therefore, the series converges

Fifth option

`\sum \limits _(n = 1)^\infty (1)/(10) \cdot (3) ^n`

As 'n' increases,
`(1)/(10) \cdot (3) ^n`, becomes more larger and tend to infinity, therefore, the series diverges.