Limit of a sequence given by (1/3)^k

[teng125]

Determine the limit of the sequence (sn)n=N given by

n(sum)k=1 (1/3)^k , n is natural numbers.
i don't understand what is the meaning
can anyonepls help...thanx

 

[HallsofIvy]

Determine the limit of the sequence (sn)n=N given by
n(sum)k=1 (1/3)^k , n is natural numbers.
i don't understand what is the meaning
can anyonepls help...thanx

Sorry, I don't understand what it means either! is either of the "n"s in "(sn)n" a subscript? What does (sn)n= N mean? what does n(sum)k mean?

 

[Architeuthis Dux]

The limit of a sequence is the value that the terms of the sequence approach as the index (in this case, k) approaches infinity. In this case, the sequence given by (1/3)^k approaches 0 as k approaches infinity, since the value of (1/3)^k decreases as k increases. Therefore, the limit of the sequence is 0.

Now, for the sequence (sn)n=N, we can rewrite it as:

s1 = (1/3)^1
s2 = (1/3)^2
s3 = (1/3)^3
...
sn = (1/3)^n

As n approaches infinity, the value of (1/3)^n approaches 0. Therefore, the limit of the sequence (sn)n=N is also 0.

In summary, the limit of the given sequence is 0, since the terms of the sequence approach 0 as the index approaches infinity. I hope this helps clarify the meaning of the limit of a sequence.