Answer to Question #205032 in Calculus for abcdefna

Question #205032

Determine whether the following series converges or diverges. If the series converges, find its sum.

a. βˆ‘βˆžπ‘›=1 (1+2n+1)/3n

b. βˆ‘βˆžπ‘›=1 (3+2n)/(2n+2)

c. βˆ‘βˆžπ‘›=1 1/(4n2-1)

1
Expert's answer
2021-06-10T07:26:58-0400

a.


βˆ‘n=1∞1+2n+13n=diverges\displaystyle\sum_{n=1}^{\infin}\dfrac{1+2^{n+1}}{3n}=\text{diverges}

lim⁑nβ†’βˆžan=lim⁑nβ†’βˆž1+2n+13n=∞=ΜΈ0\lim\limits_{n\to \infin}a_n=\lim\limits_{n\to \infin}\dfrac{1+2^{n+1}}{3n}=\infin\not=0

The series diverges by the Test for divergence.


b.


βˆ‘n=1∞3+2n2n+2=diverges\displaystyle\sum_{n=1}^{\infin}\dfrac{3+2^{n}}{2^{n+2}}=\text{diverges}

lim⁑nβ†’βˆžan=lim⁑nβ†’βˆž3+2n2n+2=lim⁑nβ†’βˆž32n+2n2n2n+22n\lim\limits_{n\to \infin}a_n=\lim\limits_{n\to \infin}\dfrac{3+2^{n}}{2^{n+2}}=\lim\limits_{n\to \infin}\dfrac{\dfrac{3}{2^{n}}+\dfrac{2^{n}}{2^{n}}}{\dfrac{2^{n+2}}{2^{n}}}

=lim⁑nβ†’βˆž32n+14=14=ΜΈ0=\lim\limits_{n\to \infin}\dfrac{\dfrac{3}{2^{n}}+1}{4}=\dfrac{1}{4}\not=0

The series diverges by the Test for divergence.


c.


βˆ‘n=1∞14n2βˆ’1=βˆ‘n=1∞1(2nβˆ’1)(2n+1)\displaystyle\sum_{n=1}^{\infin}\dfrac{1}{4n^2-1}=\displaystyle\sum_{n=1}^{\infin}\dfrac{1}{(2n-1)(2n+1)}

=12(βˆ‘n=1∞12nβˆ’1βˆ’βˆ‘n=1∞12n+1)=\dfrac{1}{2}\bigg(\displaystyle\sum_{n=1}^{\infin}\dfrac{1}{2n-1}-\displaystyle\sum_{n=1}^{\infin}\dfrac{1}{2n+1}\bigg)=12(11βˆ’13+13βˆ’15+...)=\dfrac{1}{2}\bigg(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...\bigg)

βˆ‘n=1∞14n2βˆ’1=12\displaystyle\sum_{n=1}^{\infin}\dfrac{1}{4n^2-1}=\dfrac{1}{2}



The series converges and its sum is 12.\dfrac{1}{2}.



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