推 math1209 :Apostol 裡面的題目. 請至下列網站查詢證明... 03/22 19:36
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做題目時卡住∫Σ
(第一題是有點看不懂題目)
1.Let {a_n} be a sequence of real number. For x>=0, define
[x
A(x)=Σa_n=Σa_n,
n<=x n=1
where [x] is the greatest integer in x and empty sums are interpreted
as zero. Let f have a continuous derivative in the interval 1<= x <=a.
Use Stieltjes integrals to derive the following formula:
a '
Σa_nf(n)=-∫A(x)f (x)dx+A(a)f(a).
n<=a 1
2.Let g_1(x)=x-[x]-1/2 if x=/= integer, and let g_1(x)=0 if x=integer.Also,
x '' ,
Let g_2(x)=∫g_1(t)dt. If f is continuous on [1,n] prove thst Euler s
0
summation formula implies that
n n n ''
Σf(k)=∫f(x)dx-∫g_2(x)f (x)dx+[f(1)+f(n)]/2.
k=1 1 1
3.If A遞增 on [a,b],prove that we have
_b _c _b
a) ∫ fdA=∫ fdA+∫ fdA, (a<c<b),
a a c
_b _b _b
b) ∫ (f+g)dA<=∫ fdA+∫ gdA,
a a a
b b b
c) ∫ (f+g)dA>=∫ fdA+∫ gdA,
- a - a - a
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