Write out the first five term of the sequence determi whether the sequence converges and if so find its limit
(n/(n+3)
n=1 to n = ∞
To find the first five terms of the sequence, we substitute values for n from 1 to 5 and evaluate the expression (n/(n+3)):
When n = 1: (1/(1+3)) = 1/4
When n = 2: (2/(2+3)) = 2/5
When n = 3: (3/(3+3)) = 3/6 = 1/2
When n = 4: (4/(4+3)) = 4/7
When n = 5: (5/(5+3)) = 5/8
First five terms of the sequence: 1/4, 2/5, 1/2, 4/7, 5/8
Now, let's determine whether the sequence converges and find its limit.
A sequence converges if its terms get arbitrarily close to a certain value as n approaches infinity. To test for convergence, we observe the behavior of the terms as n increases.
When n is very large (approaching infinity), the term (n/(n+3)) simplifies:
lim(n->∞) (n/(n+3)) = lim(n->∞) (1/(1+(3/n))) = 1/(1+0) = 1/1 = 1.
Therefore, the limit of the sequence is 1, indicating convergence.