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.

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.

The Correct Accurate Answer   is option  4

Step-by-step explanation:

Firstly, let us define diverge by comparing it to converge :  

a divergent series is an infinite series that is not convergent, meaning:  

the infinite sequence does not have a finite limit ,  

if a series converges, the individual terms must approach zero

if a series divergent, then the series does not have a limit  

examples of a diverge will mostly have 1,2,4,8,16,

so, option 4 is the series that diverges  ... because n will forever be squared, it will not have a limit in this since, meaning it is a divergent series .

therefore, I hope this helps!



a orange at the grocery store idk what other product that could be