There is no general algorithm for finding the sum of a series. Indeed, it is often the case that series do not converge to something "nice". Whether you can write down a "nice" form for the sum usually depends on whether you can "spot" what is happening. In this case, it is fairly easy to spot the patterns. The first thing to note is that you can decompose your series into two sum:dmitriylm said:Homework Statement
Find the sum of the series:
(1/(4^n))+ (((-1)^n)/(3^n))
from n=0 to infinity
Homework Equations
The Attempt at a Solution
I'm not overly familiar with series and am not sure how to approach this. A lot of help guides online talk about testing for divergence/convergence but how do I go about finding the sum?
Can you see that each series tends to a particular number?dmitriylm said:I see the first sequence has a pattern of 1/(4^n) and the second one has a pattern of 1/(-3(^n)). Once I see the pattern what do I do?
Hootenanny said:Can you see that each series tends to a particular number?
Yes, each term is getting progressively closer to zero - so what is happening to the sum? Perhaps it would help if you plotted the sum.dmitriylm said:I'm not clear on this, they are getting smaller so they would be getting closer to zero?
An infinite series is a sum of an infinite number of terms. It is a mathematical concept used to represent the sum of an infinite number of terms in a sequence. It is an important concept in calculus and is used to calculate the value of a function at a given point.
To determine if an infinite series converges or diverges, you can use various tests such as the ratio test, the root test, or the comparison test. These tests compare the terms of the series to known convergent or divergent series and can help determine the behavior of the series.
To find the sum of an infinite series, you can use various methods such as the geometric series formula, the telescoping series method, or the power series method. These methods use specific formulas or techniques to find the sum of the series.
Infinite series have many real-world applications, such as in physics, engineering, and economics. They are used to model and analyze continuous processes, such as the motion of objects, electrical circuits, and financial growth or decay.
One example of an infinite series is the geometric series, where each term is multiplied by a common ratio. For example, 1 + 1/2 + 1/4 + 1/8 + ... is a geometric series with a common ratio of 1/2. Another example is the harmonic series, which is the sum of the reciprocals of the positive integers (1 + 1/2 + 1/3 + 1/4 + ...).