Question

What is the greatest integer not exceeding the sum sigma (n=1 till 1599 )1/sqrt(n)​

Answer 1

Answer:

the greatest integer not exceeding the sum = 78

Step-by-step explanation:

What is the greatest integer not exceeding the sum sigma (n=1 till 1599 )1/sqrt(n)​

1/√n  = 2/2√n

= 2/(√n + √n)

2/(√n + √n) >  2/(√n + √n+1)

by rationalizing

2/(√n + √n+1)  = 2(√n+1 - √n)

=> 1/√n > 2(√n+1 - √n)

By Adding all terms from 1 to 1599 we will get

2(√1600 - 1)    (as all other terms will get cancelled)

= 2(40 - 1)

= 78

=> sum > 78

2/(√n + √n) <  2/(√n + √n-1)

by rationalizing

2/(√n + √n-1)  = 2(√n - √n-1)

By Adding all terms from 2 to 1599 we will get

2(√1599 - 1)    (as all other terms will get cancelled)

= 77.98

1/√1 + 77.98  = 78.98

=> sum < 78.98

=>   78 < Sum < 78.98

the greatest integer not exceeding the sum = 78