Ajay R. answered • 06/10/21
Experienced Tutor Specializing in both Algebra and Calculus
D
When looking at a pattern it is always crucial to identify how the pattern is changing. For example, when I see that the pattern is 1, 3, 5, 7, 9, 11 I immediately think that this pattern involves all odd numbers. Knowing the fact that any integer number multiplied by 2 is even and that any number, when divided by 2 that has a remainder of 1, can be defined as odd. We can assume that the denominator in this instance has some variable k multiplied by 2 and then subtracted by 1. To ensure that no matter what number is inputted to k, the result will always be odd.
So, the expansion of
∑ 1/(2k-1) from k = 1 to k=6 is 1/(2*1-1) + 1/(2*2-1) + 1/(2*3-1) + 1/(2*4-1) + 1/(2*5-1) + 1/(2*6-1) or
1 + 1/3 + 1/5 + 1/7 + 1/9+ 1/11.
