Yae Miteo said:Homework Statement
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
Homework Equations
[tex] \sum_{n=1}^{\infty} \frac{1 + 2^n}{3^n} [/tex]
The Attempt at a Solution
Hello,
I have tried to find the sum of this series using both the limit and integral tests, but I cannot get the right answer. My answer is 1/3, but the book says it is 5/2. How can I solve this?
A series convergence problem is a mathematical problem that involves determining whether an infinite series, or a sum of an infinite number of terms, will converge to a finite value or diverge to infinity.
To find the sum of a convergent series, you can use various methods such as the geometric series formula, telescoping series, or the ratio test. These methods involve manipulating the series and applying mathematical concepts to determine the sum.
The ratio test is a mathematical test that is used to determine the convergence or divergence of a series. It involves taking the limit of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit is equal to 1, the test is inconclusive.
No, a divergent series cannot have a finite sum. A divergent series will either diverge to infinity or oscillate between positive and negative values, but it will never converge to a finite value.
There are no shortcuts or tricks to solve a series convergence problem. It requires a solid understanding of mathematical concepts and techniques, as well as practice and patience. Additionally, using a graphing calculator or software program can help with visualizing the series and determining its convergence or divergence.