Math 2414 Sections 8 3 Infinite Series Infinite Series and the Sequence of Partial Sums From a given sequence of real or complex numbers we can always generate a new sequence by adding together successive terms Thus if the given sequence has the terms a1 a2 K an K we may form in succession the partial sums S1 a1 S2 a1 a2 S3 a1 a2 a3 and so on the partial sum Sn of the first n terms being defined as follows n S n a1 a2 K an ak k 1 Furthermore the infinite series or simply series describes the sum of an infinite set of numbers and is denoted in the following ways a1 a2 a3 K or a1 a2 K an K or a k k 1 If there is a real or complex number L such that lim Sn L n we say that the series a k 1 k is convergent and has the sum L in which case we write a k L k 1 If Sn diverges we say that the series a k 1 k diverges and has no sum In summary if you add a finite number of the terms say the first n terms and denote their sum S S by n This is called the nth partial sum of the series The sequence n generated by finding successive partial sums is extremely important It is called a sequence of partial sums and its limit is the value of the infinite series If the partial sums tend to a finite limit then the infinite series converges If the partial sums have no finite limit then the infinite series diverges 1 k k 1 Consider the infinite series k 1 a Find the first four terms of the sequence of partial sums S b Find an expression for n and make a conjecture about the value of the series Write the first four terms of the sequence of partial sums then estimate the limit of the series or state that it does not exist 1 k k k 1 Infinite Series Recall that every infinite series a k k 1 S1 a1 has a sequence of partial sums S2 a1 a2 S3 a1 a2 a3 n where in general S n ak k 1 for n 1 2 3 K To evaluate an infinite series it is necessary to determine a formula for the sequence of partial S sums n and then find its limit This procedure can be carried out with the series that we discuss in this section geometric series and telescoping series Geometric Series A geometric series is a series of the form a ar ar 2 K ar k k 0 in which a and r are fixed real numbers a is the first term of the series The first term is based on the given starting value of k r is called the ratio of the sum r can be positive or negative Neither a nor r can be 0 Theorem 8 7 Geometric Series Let r and a be real numbers If then the series diverges r 1 then ar k 0 k a r 1 converges and the sum is 1 r If Evaluate the following geometric series or state that the series diverges 9 9 9 9 2 3 K n K 100 100 100 100 Theorem 8 8 Properties of Convergent Series a ca 1 Suppose k converges to A and let c be a real number The series k converges ca c ak cA and k a b a b 2 Suppose k converges to A and k converges to B The series k k a b ak bk A B converges and k k 3 Whether a series converges does not depend on a finite number of terms added to or ak ak k 1 k M M removed from the series Specifically if is a positive integer then and both converge or both diverge However the value of a convergent series does change if nonzero terms are added or deleted Evaluate the following series or state that it diverges 1 1 k k 0 Evaluate the following series or state that they diverge 2 3 k 1 k 2k 1 1 k 3 3 k 1 k 5 2 k 1 k 1 3k k 1 1 10 3 k 2 Evaluate the following series or state that it diverges k 3 k 1 4 Write 1 035 1 0353535K as a geometric series and express its value as a fraction Telescoping Series With geometric series we carried out the entire evaluation process by finding a formula for the sequence of partial sums and evaluating the limit of the sequence Not many infinite series can be subjected to this sort of analysis With another class of series called telescoping series it can a be done An infinite series k whose terms can be expressed as a difference of the form an bn bn 1 is known as a telescoping series Informally a series is called a telescoping series if there is an internal cancellation in the partial sums Evaluate the following series 1 1 k k 1 3 3 k 1 1 k k 1 k 1 Evaluate the following series k 1 ln k k 1