Question

Show that {an} is monotonic and determine whether {an} is convergent or divergent. If the sequence is convergent, find its limit.

Answer 1

\[a) \left\{ a _{n} \right\}=\left\{ \frac{ \sqrt{n} }{ 1+\sqrt{n} } \right\}\]

Answer 2

\[b) \left\{ a _{n} \right\} = \left\{ 4+ \frac{ 1 }{ 2^{n} } \right\}\]

Answer 3

post your other question separetely

Answer 4

define a function
f(x) = sqrt(x)/(1 + sqrt(x))
do second derivative test.
or induction might be helpful to show the monotocity.

Answer 5

how do you show the monotocity?

Answer 6

how do you show that function is increasing?

Answer 7

so i would have to use the quotient rule to determine the derivative of f(x) rigt?

Answer 8

you can show that the slope of the function is increasing.

Answer 9

or you can show that via induction.

Answer 10

i would use the derivative of f(x) to show if its increasing or decreasing.

Answer 11

yes you can do that ... show that the slope is greater than zero for all x>1
for
\[f(x) = \frac{\sqrt{x}}{1 + \sqrt{x}}\]

Answer 12

so when i have my derivative, how would i be able to tell that the slope is greater for zero for all x>1?

Answer 13

well i know the denominator will always be positive because it's squared

Answer 14

and the tops have square roots in them.

Answer 15

is my sequence bounded? how do i tell?

Answer 16

just find a suitable upper bound for your sequence

Answer 17

i dont understand.

Answer 18

you can show that this sequence is always less than one.