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Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
L'Hôpital's Rule
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
Challenge Quizzes
Consider the sequence \( \left \{ a_n \right \}_{n \in \mathbb{N}} \) defined by \[\begin{align} a_1 &= \frac{3}{4} \\ a_n &= \frac{3^n}{4n} - \frac{3^{n-1}}{4 (n-1)}, n \in \mathbb{N} , n > 1 .\end{align} \] What is the value of \( \displaystyle { \sum_{n=1}^{\infty} a_n } ?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Consider the sequence \( \left \{ a_n \right \}_{n \in \mathbb{N}} \) defined by \[\begin{align} a_1 &= 3 \cot 5 \\ a_n &= \frac{3}{n} \cot \frac{5}{n} -\frac{3}{n-1} \cot \frac{5}{n-1} , n \in \mathbb{N} , n > 1 .\end{align} \] What is the value of \( \displaystyle { \sum_{n=1}^{\infty} a_n } ?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Consider the sequence \( \left \{ a_n \right \}_{n \in \mathbb{N}} \) defined by \[\begin{align} a_1 &= -\frac{2}{\log \cos 7} \\ a_n &= \frac{2}{(n-1)^2 \log \cos \frac{7}{n-1}}-\frac{2}{n^2 \log \cos \frac{7}{n}} , n \in \mathbb{N} , n > 1 .\end{align} \] What is the value of \( \displaystyle { \sum_{n=1}^{\infty} a_n } ?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Consider the sequence \( \left \{ a_n \right \}_{n \in \mathbb{N}} \) defined by \[\begin{align} a_1 &= \frac{\log 3}{ \log 6} \\ a_n &= \frac{ \log (3n)}{\log (n+5)} - \frac{\log (3(n-1))}{\log ((n-1)+ 5)}, n \in \mathbb{N} , n > 1 .\end{align} \] What is the value of \( \displaystyle { \sum_{n=1}^{\infty} a_n } ?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Consider the sequence \( \left \{ a_n \right \}_{n \in \mathbb{N}} \) defined by \[\begin{align} a_1 &= 27 \\ a_n &= {(3n)}^{\frac{3}{n}}- {(3(n-1))}^{\frac{3}{n-1}}, n \in \mathbb{N} , n > 1 .\end{align} \] What is the value of \( \displaystyle { \sum_{n=1}^{\infty} a_n } ?\)