Question

Is this correct? The answer I picked was the only one that had no limit. If I did it correctly the other choices limit was zero.

Answer 1

Step-by-step explanation:

These are all examples of p-series:

∑(1 / nᵖ), where p>0.

If p > 1, the series converges.  If 0 < p ≤ 1, the series diverges.

First option:

∑(1/n⁵)

Here, p = 5.  Since 5 > 1, the series converges.

Second option:

∑((√n+3)/n³)

∑((√n)/n³) + ∑(3/n³)

∑(1/n^2.5) + 3 ∑(1/n³)

In the first sum, p = 2.5.  In the second sum, p = 3.  Both are greater than 1, so the series converges.

Third option:

∑((n−4)/(n⁴√n))

∑(1/(n³√n)) − ∑(4/(n⁴√n))

∑(1/n^3.5) − 4 ∑(1/n^4.5)

In the first sum, p = 3.5.  In the second sum, p = 4.5.  Both are greater than 1, so the series converges.

Fourth option:

∑(1/∛n)

∑(1/n^⅓)

Here, p = ⅓.  This is less than 1, so the series diverges.

Note: if a series is converging, then the limit is 0.

However, if the limit of a series is 0, it does not necessarily mean that series is converging.

Here, the limit of all 4 options is 0.  However, the fourth option is a diverging series.