We find the interval of convergence of a power series.
Interval of Convergence
Power Series
We will find the interval of convergence of a power series.
Loosely speaking, a power series is a polynomial of infinite degree. For example, The name power series comes from the fact that we have an infinite series that contains powers of the variable . In the formal definition of a power series below, we allow for powers of rather than just powers of . The number is called the center of the power series. The power series in the example above has center at .
Some examples of power series are: The first two have center at and the third is centered at . The question we ask when confronted with a power series is, “for which values of does the power series converge?” The theory tells us that the power series will converge in an interval centered at the center of the power series. To find this interval of convergence, we frequently use the ratio test.
example 1 Find the interval of convergence of the power series .
This is a geometric series with common ratio , and hence it converges if and only in . Hence, the interval of convergence of the power series is . In other words, for any value of in this interval, the resulting series will converge and for any value of outside of this interval, the series will diverge. Notice that the interval of convergence is centered around , which is the center of the power series. Furthermore, since the series is geometric, we know that the sum of the series is for any value of in the interval of convergence. We will return to this fact in the next section.
This is a geometric series with common ratio , and hence it converges if and only in . Hence, the interval of convergence of the power series is . In other words, for any value of in this interval, the resulting series will converge and for any value of outside of this interval, the series will diverge. Notice that the interval of convergence is centered around , which is the center of the power series. Furthermore, since the series is geometric, we know that the sum of the series is for any value of in the interval of convergence. We will return to this fact in the next section.
example 2 Find the interval of convergence of the power series .
We will use the ratio test: By the rules for the ratio test, the series converges when and diverges when . Unfortunately, the ratio test gives no conclusion when , which corresponds to . To determine the behavior of the series at these values, we plug them into the power series. If , the power series becomes which is a -series with . Since , this series converges and therefore our power series converges when . Next, if , the power series becomes: which is an alternating -series and so it converges. Therefore our power series converges when . The interval of convergence of the power series is thus , and we again note that this is an interval centered about the center of the power series, .
We will use the ratio test: By the rules for the ratio test, the series converges when and diverges when . Unfortunately, the ratio test gives no conclusion when , which corresponds to . To determine the behavior of the series at these values, we plug them into the power series. If , the power series becomes which is a -series with . Since , this series converges and therefore our power series converges when . Next, if , the power series becomes: which is an alternating -series and so it converges. Therefore our power series converges when . The interval of convergence of the power series is thus , and we again note that this is an interval centered about the center of the power series, .
example 3 Find the interval of convergence of the power series .
We will use the ratio test: By the rules for the ratio test, the series converges regardless of the value of . Hence the interval of convergence is .
We will use the ratio test: By the rules for the ratio test, the series converges regardless of the value of . Hence the interval of convergence is .
example 4 Find the interval of convergence of the power series .
We will use the ratio test: By the rules for the ratio test, the series converges when and diverges when . Unfortunately, the ratio test gives no conclusion when , which corresponds to and . To determine the behavior of the series at these values, we plug them into the power series. If , the power series becomes which is the divergent harmonic series. Next, if , the power series becomes: which is the convergent alternating harmonic series. The interval of convergence of the power series is thus , and we again note that this is an interval centered about the center of the power series, .
We will use the ratio test: By the rules for the ratio test, the series converges when and diverges when . Unfortunately, the ratio test gives no conclusion when , which corresponds to and . To determine the behavior of the series at these values, we plug them into the power series. If , the power series becomes which is the divergent harmonic series. Next, if , the power series becomes: which is the convergent alternating harmonic series. The interval of convergence of the power series is thus , and we again note that this is an interval centered about the center of the power series, .
example 5 Find the interval of convergence of the power series .
We will use the ratio test: By the rules for the ratio test, the series converges when and diverges when . Unfortunately, the ratio test gives no conclusion when , which corresponds to and . To determine the behavior of the series at these values, we plug them into the power series. If , the power series becomes which diverges by the limit comparison test with the harmonic series. Next, if , the power series becomes: which converges by the alternating series test. The interval of convergence of the power series is thus , and we again note that this is an interval centered about the center of the power series, .
We will use the ratio test: By the rules for the ratio test, the series converges when and diverges when . Unfortunately, the ratio test gives no conclusion when , which corresponds to and . To determine the behavior of the series at these values, we plug them into the power series. If , the power series becomes which diverges by the limit comparison test with the harmonic series. Next, if , the power series becomes: which converges by the alternating series test. The interval of convergence of the power series is thus , and we again note that this is an interval centered about the center of the power series, .
Here is a detailed, lecture style video on the interval of convergence of a power series:
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