How do you find a formula for the sum n

Priscilla Johnston 2021-12-13 Answered
How do you find a formula for the sum n terms i=1n(1+in)(2n) and then find the limit as n
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Expert Answer

Ben Owens
Answered 2021-12-14 Author has 27 answers

i=1n(1+in)(2n)=3n+1n
limni=1n(1+in)(2n)=3
Explanation
Let Sn=i=1n(1+in)(2n)
Sn=i=1n(2n+2in2)
Sn=2ni=1n(1)+2in2i=1n(i)
And using the standard results: i=1nr=12n(n+1)
We have:
Sn=2n(n)=2n212n(n+1)
Sn=2+n+1n
Sn=(2n)+(n+1)n
Sn=3n+1n
Now we examine the behaviour of Sn as n
We have:
Sn=3n+1n
Sn=3+1n
limnSn=limn{3+1n}
limnSn=limn(3)limn(1n)
And as 1n0 as n we have;
limn
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Gerald Lopez
Answered 2021-12-15 Author has 29 answers

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