Do the following diverge or converge? explain why.
Question: Do the following diverge or converge? explain why.
In this blog post, we will explore the concept of divergence and convergence of sequences and series. We will also see some examples and methods to determine whether a given sequence or series diverges or converges.
A sequence is a list of numbers that follows a certain rule or pattern. For example, the sequence 1, 2, 4, 8, 16, ... is a sequence where each term is double the previous one. A series is the sum of the terms of a sequence. For example, the series 1 + 2 + 4 + 8 + 16 + ... is the sum of the terms of the previous sequence.
A sequence or a series can either diverge or converge. Divergence means that the terms of the sequence or the series get larger and larger without bound, or oscillate between positive and negative values without settling down. Convergence means that the terms of the sequence or the series get closer and closer to a fixed value, called the limit.
To determine whether a sequence or a series diverges or converges, we can use various tests and criteria, depending on the type and form of the sequence or series. Some common tests are:
- The nth term test: If the limit of the nth term of a series is not zero, then the series diverges.
- The geometric series test: A geometric series is a series where each term is a constant multiple of the previous one. For example, 1 + 2 + 4 + 8 + ... is a geometric series with a common ratio of 2. A geometric series converges if and only if the absolute value of the common ratio is less than 1.
- The integral test: If a series is composed of positive, decreasing terms that are also terms of a continuous function f(x), then the series converges if and only if the improper integral from 1 to infinity of f(x) dx converges.
- The comparison test: If a series is composed of positive terms, and there is another series with positive terms that converges, such that each term of the original series is less than or equal to the corresponding term of the other series, then the original series also converges. Conversely, if there is another series with positive terms that diverges, such that each term of the original series is greater than or equal to the corresponding term of the other series, then the original series also diverges.
- The ratio test: If a series has nonzero terms, and there is a positive number r such that for all sufficiently large n, the absolute value of the ratio of the (n+1)th term to the nth term is less than r, then the series converges. If this ratio is greater than or equal to 1 for all sufficiently large n, then the series diverges.
- The root test: If a series has nonzero terms, and there is a positive number r such that for all sufficiently large n, the nth root of the absolute value of the nth term is less than r, then the series converges. If this root is greater than or equal to 1 for all sufficiently large n, then the series diverges.
These are some of the most common tests for divergence and convergence, but there are many more. It is important to understand how to apply them correctly and interpret their results. In some cases, more than one test can be used, and in some cases, none of these tests may work. In those situations, we may need to use other techniques or tools to find out whether a sequence or a series diverges or converges.
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