Question

I need help.. i really want to go sleep.. thank you so much...

** the two questions are boxed in red**

Answer 1

1) True 2) False

Explanation:

1) Given
`\sum\limits_(k=0)^8(1)/(k+3)=\sum\limits_(i=3)^(11)(1)/(i)`

To verify that the above equality is true or false:

Now find
`\sum\limits_(k=0)^8(1)/(k+3)`

Expanding the summation we get

`\sum\limits_(k=0)^8(1)/(k+3)=(1)/(0+3)+(1)/(1+3)+(1)/(2+3)+(1)/(3+3)+(1)/(4+3)+(1)/(5+3)+(1)/(6+3)+(1)/(7+3)+(1)/(8+3)`
`\sum\limits_(k=0)^8(1)/(k+3)=(1)/(3)+(1)/(4)+(1)/(5)+(1)/(6)+(1)/(7)+(1)/(8)+(1)/(9)+(1)/(10)+(1)/(11)`

Now find
`\sum\limits_(i=3)^(11)(1)/(i)`

Expanding the summation we get

`\sum\limits_(i=3)^(11)(1)/(i)=(1)/(3)+(1)/(4)+(1)/(5)+(1)/(6)+(1)/(7)+(1)/(8)+(1)/(9)+(1)/(10)+(1)/(11)`

Comparing the two series we get,

`\sum\limits_(k=0)^8(1)/(k+3)=\sum\limits_(i=3)^(11)(1)/(i)` so the given equality is true.

2) Given
`\sum\limits_(k=0)^4(3k+3)/(k+6)=\sum\limits_(i=1)^3(3i)/(i+5)`

Verify the above equality is true or false

Now find
`\sum\limits_(k=0)^4(3k+3)/(k+6)`

Expanding the summation we get

`\sum\limits_(k=0)^4(3k+3)/(k+6)=(3(0)+3)/(0+6)+(3(1)+3)/(1+6)+(3(2)+3)/(2+6)+(3(3)+4)/(3+6)+(3(4)+3)/(4+6)`

`\sum\limits_(k=0)^4(3k+3)/(k+6)=(3)/(6)+(6)/(7)+(9)/(8)+(12)/(8)+(15)/(10)`

now find
`\sum\limits_(i=1)^3(3i)/(i+5)`

Expanding the summation we get

`\sum\limits_(i=1)^3(3i)/(i+5)=(3(0))/(0+5)+(3(1))/(1+5)+(3(2))/(2+5)+(3(3))/(3+5)`

`\sum\limits_(i=1)^3(3i)/(i+5)=(3)/(6)+(6)/(7)+(9)/(8)`

Comparing the series we get that the given equality is false.

ie,
`\sum\limits_(k=0)^4(3k+3)/(k+6)\\eq\sum\limits_(i=1)^3(3i)/(i+5)`