→ AbiM:你沒有照題號吧 有一個四小題 部是只有三小題 01/27 13:21
上學期期末考
1.Let f:[1,∞)→R be nonnegative and decreasing and let
n n
dn = Σ f(k) - ∫f(x)dx , n belong to N
k=1 1
Prove that {dn}n belong to N converges.
2.Test the following series for absoulte convergence ,conditional
convergence , or divergence.
(1) ∞ 1 (2) ∞ n^100
Σ ───── Σ ────
n=1 n^(2/3) n=1 n!
(3)∞ 3 1 2 3 1 2
Σ ── - ── - ── + ── - ── - ── + ...
n=1 √1 √2 √3 √4 √5 √6
3. x‧n^(3/2)
Let fn(x) = ────── , n belong to N
1 + x‧n^2
Prove that {fn}n belong to N converges pointwise on [0,∞) and
determine whether or not the convergence is uniform on [0,∞).
How about on [0.001,∞)?
4.Let a,b belong to R with a < b and let fn be integrable on [a,b]
for all n belong to N .Prove that if {fn}n belong to N converges
uniformly on [a,b], then f is integrable on [a,b] .
5.(1)State the Weierstrass M-test .
(2)Given a real-valued function that is continuous everywhere on
R but differentiable nowhere on R .
6.(1)Find the Fourier series for the tunction f(x) = │x│, x belong
to[-π,π]
(2) ∞ 1 π
Prove that Σ ───── = ──
n=1 (2n-1)^2 8
7.Suppose that Sn is the n-th partial sum of the Fourier series of an
integrable function f with period 2π .Prove that
1 π 1 π 1 π
Sn(x) = ──‧∫f(t)Dn(x-t)dt = ──∫f(x-t)Dn(t)dt + ──∫f(x-t)Dn(t)dt
π -π π 0 π 0
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