Limit of a sequence

[courteous]

What is the limit of the following series: \(\displaystyle \frac{1}{7},\frac{2}{12},\frac{3}{17},\frac{4}{22},\frac{5}{27},...\)

I've gotten the quotient \(\displaystyle q=\frac{7}{6}\) and as \(\displaystyle q>1\), shouldn't the series be disconvergent (not convergent), thus having no \(\displaystyle \lim\) :?:

 

[stapel]

What is the limit of the following series: \(\displaystyle \frac{1}{7},\frac{2}{12},\frac{3}{17},\frac{4}{22},\frac{5}{27},...\)

I've gotten the quotient \(\displaystyle q=\frac{7}{6}\) and as \(\displaystyle q>1\), shouldn't the series be disconvergent (not convergent), thus having no \(\displaystyle \lim\) :?:

On what basis did you conclude that this was geometric, or that this was a series rather than a sequence?

. . . . .(2/12)/(1/7) = (1/6)(7/1) = 7/6 = 1.16666...

. . . . .(3/17)/(2/12) = (3/17)(6/1) = 18/17 = 1.0588235...

. . . . .(4/22)/(3/17) = (2/11)(17/3) = 34/33 = 1.030303...

. . . . .(5/27)/(4/22) = (5/27)(11/2) = 55/54 = 1.0185185185...

Instead, try finding a pattern that works for more than just the first two terms. :wink:

Eliz.

 

[courteous]

stapel, thank you leading me back to the right path. I have presumed (that's the right word :lol: ; def.: to undertake without leave or clear justification) a geometric series (it's also not a series ). So, there are arithmetic sequence, geometric sequence and, what, "sequence" (as all other kinds of sequences)?
Yes, that I can solve easily. (Is there any comma missing in the previous sentence?)

Should anyone be interested in solution, I'll be glad to post it.

 

[PAULK]

What is the limit of the following series: \(\displaystyle \frac{1}{7},\frac{2}{12},\frac{3}{17},\frac{4}{22},\frac{5}{27},...\)

I've gotten the quotient \(\displaystyle q=\frac{7}{6}\) and as \(\displaystyle q>1\), shouldn't the series be disconvergent (not convergent), thus having no \(\displaystyle \lim\) :?:

Assuming this SEQUENCE (not a series -- a series is a sum of a sequence.) has the obvious pattern -- that is, the tops are just the integers, n, and the bottom are the arithmetic sequence: 5n + 2, you want

lim n/(5n + 2)
n->inf

That limit would appear to be obviously 1/5.

What's that,you say -- it's not obvious? Try this:

n 1/n
------- ---- =
5n + 2 1/n

1
-------
5 + 2/n

Now the limit of that is
1
----- = 1/5
5 + 0