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.
