Question

How to find the nth term for this 2 series:

please tell me with steps by steps with formula included.Thanks

(a) 1-2+3-4+5-.....

(b) 1/2+2/3+3/4+4/5+...

Answer 1

a)

The sum of the terms will depend on the value of n if even or odd,

case I: n is even,

since n is even,

     

Adding the above two we get,

there are even number of +1,-1 pairs hence,

Case II: n is odd,

similar to the above procedure, you will get

b)

in this case,

Here, the second is the sum of a harmonic progression and the sum till n terms of an harmonic progression is approximated by the middle reimann sum and is given as,

where a is the denominator of the first term and d is common difference between two consecutive denominators.