Question

determine the nature of Series sigma n=1 to infinity


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Answer 1

Answer:

Given the sequence  {an}={1/2n}=1/2, 1/4, 1/8, … , consider the following sums:

a1a1+a2a1+a2+a3a1+a2+a3+a4====1/21/2+1/41/2+1/4+1/81/2+1/4+1/8+1/16====1/23/47/815/16(8.2.1)

In general, we can show that

a1+a2+a3+⋯+an=2n−12n=1−12n.(8.2.2)

Let  Sn  be the sum of the first  n  terms of the sequence  {1/2n} . From the above, we see that  S1=1/2 ,  S2=3/4 , etc. Our formula at the end shows that  Sn=1−1/2n .

Now consider the following limit:

limn→∞Sn=limn→∞(1−1/2n)=1.(8.2.3)

This limit can be interpreted as saying something amazing: the sum of all the terms of the sequence  {1/2n}  is 1.} This example illustrates some interesting concepts that we explore in this section. We begin this exploration with some definitions.

Step-by-step explanation: