# Yes, Use P-Series Test to Confirm Series Convergence

• naspek
In summary, you can use the p-series test to confirm that the series \sum_{n = 0}^{\infty} \frac{1}{(n + 1)^2} will converge, where p = 2. This is because the p-series test states that a series will converge if p > 1, and in this case, p = 2. You can also use the comparison test, but it is not necessary if you already know the p-series test.
naspek
if my an = 1 / (n+1)^2
can i use p series test to confirm that the series will converge..
where p = 2..

Sure. If your series is something like this,
$$\sum_{n = 0}^{\infty} \frac{1}{(n + 1)^2}$$

you can change the series index to write the series this way:
$$\sum_{n = 1}^{\infty} \frac{1}{n^2}$$

it's comparison test right?
the series will converge..
am i right?

p-series converge for p>1, so yes, your summation converges.

naspek said:
it's comparison test right?
the series will converge..
am i right?
You don't need the comparison test if you know the p-series test, and this is a p-series.

I believe that naspek was saying that he can argue that $1/(n+1)^2< 1/n^2$ and so use the comparison test without using your idea of changing the index.

## 1. What is the P-Series Test?

The P-Series Test is a convergence test used to determine if an infinite series, or a series with an infinite number of terms, converges or diverges. It is based on the power series, which is a type of infinite series where each term is a polynomial raised to a power.

## 2. How is the P-Series Test used to confirm series convergence?

The P-Series Test states that if the power series, represented as an = 1/np, has a power p greater than 1, then the series converges. Therefore, if the series being tested matches this form and p is greater than 1, it can be concluded that the series converges.

## 3. What happens if the power p is less than or equal to 1?

If p is less than or equal to 1, the P-Series Test cannot be used to determine convergence. In this case, other convergence tests such as the Comparison Test or the Integral Test may be used to determine the convergence or divergence of the series.

## 4. Can the P-Series Test be used for series with negative terms?

Yes, the P-Series Test can be used for series with negative terms as long as the power p is greater than 1. This is because the negative sign will be cancelled out when raised to a power, and the series will still follow the form of an = 1/np.

## 5. Are there any limitations to using the P-Series Test?

Yes, the P-Series Test can only be used for series that follow the form of an = 1/np. It cannot be used for other types of series such as geometric series or alternating series. Additionally, it is only applicable for determining convergence and cannot be used to determine the actual sum of the series.

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