The document discusses the p-series test and provides an example of using it to test the convergence of a series. 1) The p-series test states that the series Σ1/np converges if p>1 and diverges if p≤1. 2) As an example, it tests the series Σ(n+1)/(n+2)2 by comparing it to the divergent p-series Σ1/n, showing their limits are equal so the original series must also diverge.

2. P-SERIES OR HYPERHARMONIC TEST:
Theorem:- The infinite series
∑
1
𝑛 𝑝 =
1
1 𝑝+
1
2 𝑝+
1
3 𝑝+ …… is
convergent if p>1 and
divergent if p≤ 1.

3. Example:- test the series 𝑛=1
∞ 𝑛+1
(𝑛+2)2.
Let ∑un= 𝑛=1
∞ 𝑛+1
(𝑛+2)2 be a given series, where
un=
𝑛+1
(𝑛+2)2 is the nth term of the series. Choose an
auxiliary series ∑ vn, such that vn=
𝑛
𝑛2=
1
𝑛
.

4. Then, lim
𝑛→∞
𝑢 𝑛
𝑣 𝑛
=, lim
𝑛→∞
𝑛+1
(𝑛+2)2
1
𝑛
=, lim
𝑛→∞
𝑛+1
(𝑛+2)2 . 𝑛= lim
𝑛→∞
𝑛2 (1+
1
𝑛
)
𝑛2 (1+
2
𝑛
)2
= lim
𝑛→∞
(1+
1
𝑛
)
(1+
2
𝑛
)2
=
1+0
1+0
= 1≠0, a finite number.
Now, ∑vn=∑
1
𝑛
is divergent {∑
1
𝑛
𝑖𝑠 𝑝 − 𝑠𝑒𝑟𝑖𝑒𝑠 𝑤𝑖𝑡ℎ 𝑝 =
1 𝑖𝑛
1
𝑛 𝑝 }
Hence by comparison test, the given series
∑un= 𝑛=1
∞ 𝑛+1
(𝑛+2)2 is divergent.

5. THANK YOU