[Journal of mathematics at home University] No. 252nd postgraduate exam of Mathematics Analysis advanced algebra of Beijing Normal University in 2009

1. Evaluate $ \ DPS {\ iint_d | x | \ RD x \ RD y }$, where $ d $ is a triangle $ \ lap ABC: \ ), B (1, 1), C (2, 3) $.

 

2 convert $ \ DPS {\ iiint_vf (x, y, z) \ RD x \ rd y \ RD z} $ into multiple credits, where $ f (x, y, z) $ is a continuous function, $ V $ is a triangle: $ P (, 0), A (-, 0), B (, 2), C (, 3) $.

 

3. Calculate $ \ DPS {\ lim _ {n \ To \ infty} \ frac {1 ^ P + 2 ^ P + \ cdots + N ^ p} {n ^ p }- \ frac {n} {p + 1 }}$.

 

4. Set the function $ F $ to export everywhere. It proves that if $ F' $ has a breakpoint, it must be a second type of break point.

 

5 set $ \ sed {a_n} $ as an actual sequence, $ \ Bex s_n = A_1 + A_2 + \ cdots + a_n, \ quad \ sigma_n = \ frac {1} {n + 1} (S_1 + S_2 + \ cdots + s_n ). \ EEx $ known series $ \ DPS {\ sum _ {n = 1} ^ \ infty | s_n-\ sigma_n | ^ 2} $ convergence, proof: $ \ DPS {\ sum _ {n = 1} ^ \ infty a_n} $ convergence.

 

6. Authentication: $ \ Bex \ sum _ {n = 1} ^ \ infty \ lim _ {x \ To \ infty} \ cos \ frac {x} {2 ^ n} = \ ln \ frac {\ SiN x} {x }, \ quad \ sum _ {n = 1} ^ \ infty \ frac {1} {2 ^ n} \ tan \ frac {x} {2 ^ n} = \ frac {1} {x}-\ cot X. \ EEx $

 

7 known $ F $ continuous, $ \ DPS {\ lim _ {T \ to x} f (t) = f (x)} $. proof: $ F $ Riann can accumulate.

 

8. Set $ A $ to $ m \ times N $ matrix, $ n \ times M $ matrix $ B $ makes $ AB = e_m $ A SUFFICIENT CONDITION $ \ rank (A) = M $.

 

9 set $ A $ to a third-order matrix, $ \ rank (A) = 2 $, with a double feature value $ \ lambda_1 = \ lambda_2 = 6 $, and linear independent feature vectors belonging to $ \ lambda_1 = \ lambda_2 = 6 $ are $ \ Bex \ alpha_1 = (, 0) ^ t, \ quad \ alpha_2 = (2, 1, 1) ^ t. \ EEx $ evaluate the matrix $ A $.

 

10 it is known that $ \ Sigma $ is a symmetric transformation, $ V $ is a space, and $ W $ is a subspace of $ V $. test evidence: $ W $ is the constant sub-space of $ \ Sigma $.

 

11 known $ A, B $ for complex matrix, $ A ^ {N-2} = B ^ {N-2} \ NEQ 0 $, $ A ^ {n-1} = B ^ {n-1} = 0 $. proof: $ A, B $ is similar.