Question
The series
∑
n
=
0
∞
(
−
1
)
n
(
n
+
2
)
4
n
4
+
7
n=0
∑
∞
4n ^4
+7
(−1)
n
(n+2)
is
∑
n
=
0
∞
(
−
1
)
n
(
n
+
2
)
4
n
4
+
7
n=0
∑
∞
4n ^4
+7
(−1)
n
(n+2)
is
Answers
Answered by
GPT-4o mini
To determine the convergence or the closed form of the series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (n+2)}{4^n (4n + 7)},
\]
we first rewrite the series a bit more clearly:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (n+2)}{4^n (4n + 7)}.
\]
We can investigate the convergence of this alternating series using the Alternating Series Test and compute its value if it converges.
1. **Identify the general term \( a_n \):**
\[
a_n = \frac{(n+2)}{4^n (4n + 7)}.
\]
2. **Check for convergence:**
We examine the limit of \( a_n \) as \( n \to \infty \):
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{(n+2)}{4^n(4n + 7)} = 0.
\]
This is a necessary condition for convergence of an alternating series.
3. **Check if \( a_n \) is decreasing:**
To check if \( a_n \) is decreasing, we can examine the ratio \( \frac{a_{n+1}}{a_n} \):
\[
a_{n+1} = \frac{(n+3)}{4^{n+1}(4(n+1) + 7)} = \frac{(n+3)}{4(4n + 11)} \cdot a_n.
\]
We want \( \frac{a_{n+1}}{a_n} \):
\[
\frac{a_{n+1}}{a_n} = \frac{(n+3)}{4(4n + 11)}.
\]
Now, we examine the limit of this ratio as \( n \to \infty \):
\[
\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{(n+3)}{4(4n + 11)} = \frac{1}{16} < 1.
\]
Thus, \( a_n \) is decreasing for large \( n \).
From the Alternating Series Test, since both conditions are satisfied (the limit goes to 0 and the series is decreasing), the series converges.
4. **Finding the closed form or value of the series:**
Finding an explicit closed form of the series can involve more advanced methods (e.g. generating functions or special functions) and likely does not yield a simple expression. However, its convergence guarantees that we can compute a numeric approximation if necessary.
To summarize, the series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (n+2)}{4^n (4n + 7)}
\]
is convergent. However, without further computational tools or expansions, we may not find a straightforward closed form or numeric evaluation directly from basic series analysis. Further work may involve numeric evaluations or integral representations for approximations.
If you need a numerical approximation, further methods could entail using computational techniques.
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (n+2)}{4^n (4n + 7)},
\]
we first rewrite the series a bit more clearly:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (n+2)}{4^n (4n + 7)}.
\]
We can investigate the convergence of this alternating series using the Alternating Series Test and compute its value if it converges.
1. **Identify the general term \( a_n \):**
\[
a_n = \frac{(n+2)}{4^n (4n + 7)}.
\]
2. **Check for convergence:**
We examine the limit of \( a_n \) as \( n \to \infty \):
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{(n+2)}{4^n(4n + 7)} = 0.
\]
This is a necessary condition for convergence of an alternating series.
3. **Check if \( a_n \) is decreasing:**
To check if \( a_n \) is decreasing, we can examine the ratio \( \frac{a_{n+1}}{a_n} \):
\[
a_{n+1} = \frac{(n+3)}{4^{n+1}(4(n+1) + 7)} = \frac{(n+3)}{4(4n + 11)} \cdot a_n.
\]
We want \( \frac{a_{n+1}}{a_n} \):
\[
\frac{a_{n+1}}{a_n} = \frac{(n+3)}{4(4n + 11)}.
\]
Now, we examine the limit of this ratio as \( n \to \infty \):
\[
\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{(n+3)}{4(4n + 11)} = \frac{1}{16} < 1.
\]
Thus, \( a_n \) is decreasing for large \( n \).
From the Alternating Series Test, since both conditions are satisfied (the limit goes to 0 and the series is decreasing), the series converges.
4. **Finding the closed form or value of the series:**
Finding an explicit closed form of the series can involve more advanced methods (e.g. generating functions or special functions) and likely does not yield a simple expression. However, its convergence guarantees that we can compute a numeric approximation if necessary.
To summarize, the series
\[
\sum_{n=0}^{\infty} \frac{(-1)^n (n+2)}{4^n (4n + 7)}
\]
is convergent. However, without further computational tools or expansions, we may not find a straightforward closed form or numeric evaluation directly from basic series analysis. Further work may involve numeric evaluations or integral representations for approximations.
If you need a numerical approximation, further methods could entail using computational techniques.