Calculus

L'Hôpital's Rule

L'Hopital's Rule - Convergence of Series

         

Consider the sequence {an}nN \left \{ a_n \right \}_{n \in \mathbb{N}} defined by a1=34an=3n4n3n14(n1),nN,n>1.\begin{aligned} a_1 &= \frac{3}{4} \\ a_n &= \frac{3^n}{4n} - \frac{3^{n-1}}{4 (n-1)}, n \in \mathbb{N} , n > 1 .\end{aligned} What is the value of n=1an? \displaystyle { \sum_{n=1}^{\infty} a_n } ?

Consider the sequence {an}nN \left \{ a_n \right \}_{n \in \mathbb{N}} defined by a1=3cot5an=3ncot5n3n1cot5n1,nN,n>1.\begin{aligned} a_1 &= 3 \cot 5 \\ a_n &= \frac{3}{n} \cot \frac{5}{n} -\frac{3}{n-1} \cot \frac{5}{n-1} , n \in \mathbb{N} , n > 1 .\end{aligned} What is the value of n=1an? \displaystyle { \sum_{n=1}^{\infty} a_n } ?

Consider the sequence {an}nN \left \{ a_n \right \}_{n \in \mathbb{N}} defined by a1=2logcos7an=2(n1)2logcos7n12n2logcos7n,nN,n>1.\begin{aligned} a_1 &= -\frac{2}{\log \cos 7} \\ a_n &= \frac{2}{(n-1)^2 \log \cos \frac{7}{n-1}}-\frac{2}{n^2 \log \cos \frac{7}{n}} , n \in \mathbb{N} , n > 1 .\end{aligned} What is the value of n=1an? \displaystyle { \sum_{n=1}^{\infty} a_n } ?

Consider the sequence {an}nN \left \{ a_n \right \}_{n \in \mathbb{N}} defined by a1=log3log6an=log(3n)log(n+5)log(3(n1))log((n1)+5),nN,n>1.\begin{aligned} a_1 &= \frac{\log 3}{ \log 6} \\ a_n &= \frac{ \log (3n)}{\log (n+5)} - \frac{\log (3(n-1))}{\log ((n-1)+ 5)}, n \in \mathbb{N} , n > 1 .\end{aligned} What is the value of n=1an? \displaystyle { \sum_{n=1}^{\infty} a_n } ?

Consider the sequence {an}nN \left \{ a_n \right \}_{n \in \mathbb{N}} defined by a1=27an=(3n)3n(3(n1))3n1,nN,n>1.\begin{aligned} a_1 &= 27 \\ a_n &= {(3n)}^{\frac{3}{n}}- {(3(n-1))}^{\frac{3}{n-1}}, n \in \mathbb{N} , n > 1 .\end{aligned} What is the value of n=1an? \displaystyle { \sum_{n=1}^{\infty} a_n } ?

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