Find the sum of the infinite series whose sequence of partial sums, Sn, is S sub n equals= 10-1/(n+1)
a) 10
b) 0
c) 9.99
d) Sum does not exist
actually, if the sequence of partial sums is Sn = 10 - 1/(n+1)
Sn -> 10 as n->∞
Note that we are not talking about the terms Tn of the sequence itself.
To find the sum of the infinite series, let's start by expressing the sequence of partial sums, Sn, in terms of n:
Sn = 10 - 1/(n + 1)
The sequence of partial sums represents the sum of the terms up to the nth term. We need to find the limit as n approaches infinity of Sn. Let's calculate it:
lim(n→∞) Sn = lim(n→∞) (10 - 1/(n + 1))
The second term, 1/(n + 1), becomes smaller and tends to zero as n approaches infinity. Therefore:
lim(n→∞) Sn = lim(n→∞) 10
The limit of the constant term 10, as n approaches infinity, is still 10. Thus, the sum of the infinite series is 10.
Therefore, the answer is (a) 10.
To find the sum of the infinite series, we need to determine the limit of the sequence of partial sums, Sn, as n approaches infinity. Let's break down the given formula S sub n = 10 - 1/(n + 1) to understand its pattern.
The given formula suggests that the sequence starts with 10 and subtracts 1 divided by (n + 1) for each term. As n increases, the value of 1/(n + 1) approaches zero, which means that the subsequent terms will become smaller and smaller.
To find the sum, we can take the limit of Sn as n approaches infinity:
lim(n→∞) Sn = lim(n→∞) (10 - 1/(n + 1))
Since the last term of the series becomes negligible as n approaches infinity, we can disregard it in our calculation:
lim(n→∞) Sn ≈ lim(n→∞) 10
The limit of 10 as n approaches infinity is still 10. Therefore, the sum of the infinite series is 10.
So, the answer is option a) 10.
form the first few terms as:
(10 - 1/2) + (10 - 1/3) + (10 - 1/4) + .....
= 19/2 + 29/3 + 39/4 + .....
I hope you can see that Sal Khan's video fits nicely into your problem
and that you find it quite interesting.
https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-6/v/harmonic-series-divergent
btw, I did a computer simulation and found the following:
Sum(100) = 19/2 +29/3 + 39/4 + ... + 109/101 = 4.197279
sum(500) = .... = 5.79482
sum(2000) = ... = 7.178871
sum(5000) = ... = 8.094717
sum(50,000) = 10.39695 , mmmhhhh?
so , what do you think?