MathDB
Problems
Contests
National and Regional Contests
Japan Contests
Today's Calculation Of Integral
2012 Today's Calculation Of Integral
855
855
Part of
2012 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 855
Source:
11/17/2012
Let
f
(
x
)
f(x)
f
(
x
)
be a function which is differentiable twice and
f
′
′
(
x
)
>
0
f''(x)>0
f
′′
(
x
)
>
0
on
[
0
,
1
]
[0,\ 1]
[
0
,
1
]
.For a positive integer
n
n
n
, find
lim
n
→
∞
n
{
∫
0
1
f
(
x
)
d
x
−
1
n
∑
k
=
0
n
−
1
f
(
k
n
)
}
.
\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.
lim
n
→
∞
n
{
∫
0
1
f
(
x
)
d
x
−
n
1
∑
k
=
0
n
−
1
f
(
n
k
)
}
.
calculus
integration
function
limit
calculus computations