证明: 将收敛级数 [tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrMk2INBod3J4j6UNzA1luB4=[/tex]相邻的奇偶项交换位置得到的新级数也收敛，且和不变.

2022-06-14

证明: 将收敛级数 [tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrMk2INBod3J4j6UNzA1luB4=[/tex]相邻的奇偶项交换位置得到的新级数也收敛，且和不变.

答案：

证明     设级数 [tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrMk2INBod3J4j6UNzA1luB4=[/tex]的部分和 [tex=4.786x3.286]QeAQ0AEILcMBNB5tpI7n6AiNn6grMvgBxUmO/iLlycA=[/tex] 已知级数[tex=2.571x3.286]blMv6qYIfB4oD7O5bXfToHoU4vJOCRyno82Zyr6SEmk=[/tex]收敛, 即部分和数列 [tex=2.071x1.357]zwmW4rSOVYrEAa6w5bAbhjiMVj+8XrHWvkmEp43+cO8=[/tex]收敛[tex=5.214x1.714]Vc7UJ8C4LSf3poBB8AyThiMmMIIYUNRRd1IJdGFjjO1JdxbnnkXFbGATdKgc8dEE[/tex] 同时有 [tex=4.5x1.714]OqU0SQaVHd2x+OGLCy0gvatsU7KHTyxR18qBsaLaMzBgQq4zUDCPhOA2gmizgCc/[/tex]将级数[tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrMk2INBod3J4j6UNzA1luB4=[/tex] 相邻的奇偶项交换位置后所得级数记为 [tex=2.429x3.286]3PXegz5bAQsuTODB0U8KrJlVSAqg4KI4iEVYZNjcRXEBx9PJWiyiRnzMvz2Bes/6[/tex],记级数[tex=2.429x3.286]3PXegz5bAQsuTODB0U8KrJlVSAqg4KI4iEVYZNjcRXEBx9PJWiyiRnzMvz2Bes/6[/tex]的部分和为[tex=0.929x1.429]oucPlfTdXLT39q4V6fqNczqREYzYiIrnWFj4PKuGU4I=[/tex] 考虑该级数的部分和 [tex=1.929x1.429]sQGMV01ACnsUGDB5fg7swYBjAt2w+EIyoz9qqqUn+hU=[/tex] 发现，当[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]为偶数时, 即 [tex=2.429x1.0]hwZpeUsiXv4dUA3MkAbM1w==[/tex] 时[tex=4.929x1.286]ipPfVQZghW5gTvzRnk1arph520aUOerD0tTEYF1Oc/jfJMQJsrp8GowffhLKdUPc[/tex]由于[tex=4.429x1.714]OqU0SQaVHd2x+OGLCy0gvcyyoQ/ZBl4yw/0xxPH+MLwKB4WDJqxRFtHQh8uUNOas[/tex] 可得 [tex=5.571x1.571]dN+ZbiiEc2FgivmlTSb6y6uwMWStuQsSFgR9n5Mzk2IvlQfJdZ+ooz2BngiVj/ZQPLku7XyXEb3RvrlkRzR+lw==[/tex];当[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]为奇数时,即 [tex=3.714x1.143]OKoRX8zUZ2sjFIF5Ykx73w==[/tex]时[tex=25.5x1.571]zgq5mVWfWgrS/VBA6ho5vqr0kHvu7ub8Nup4heqgeuwwcB4ke0SFyf59EyHwJwJTMWVuINwpl6gLMND+MfQ1kTaosUO+l5yXgt7S/uNPXbHQqItoOB0kW0W2eNWc+OeDv7GUayEnicd4mY4VNTqIlvxz1Ni7tSm8mOzWCJfrhtDNQhOQiRxzToKPFSVM1kQEWUb7kv3Do6uusf/GxbDqJQ==[/tex]即得 [tex=4.286x1.857]OqU0SQaVHd2x+OGLCy0gvcyyoQ/ZBl4yw/0xxPH+MLxosKf29t38MVXhg0b5z9/Y[/tex]， 故新级数[tex=2.429x3.286]3PXegz5bAQsuTODB0U8KrJlVSAqg4KI4iEVYZNjcRXEBx9PJWiyiRnzMvz2Bes/6[/tex]也收敛，且和不变.

举一反三

内容

0
输出以下4*5的矩阵。 1 2 3 4 5 2 4 6 8 10 3 6 9 12 15 4 8 12 16 20 #include int main() { int i,j,n=0; for(i=1;i<=4;i++) for(j=1; (1) ; j++, (2) ) { if( (3) ) printf(" "); printf("%d ",i*j); } printf(" "); return 0; }
1
【1】求级数X^n/n^3的收敛域【2】求级数(2^n/n+1)*x^n的收敛半径
2
设级数[tex=2.643x3.286]3PXegz5bAQsuTODB0U8KrBH+wjGZ4B2NqYZKoXzdWWo=[/tex]收敛，则必收敛的级数为未知类型：{'options': ['[tex=5.357x3.286]3PXegz5bAQsuTODB0U8KrFS1UdlKd89Wogp28nOb3ldWlp2uaeJY0TV5oliGQldx[/tex]', '[tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrOKIndDjeIVEY+UL3wsY2kM=[/tex]', '[tex=6.786x3.286]3PXegz5bAQsuTODB0U8KrJ7wkjrgx/DWuodQdMTXW1ZtqgSDTn+zvnS0RjNNqiMXRN46kkEyzpXwr7DtMhXS1Q==[/tex]', '[tex=6.071x3.286]3PXegz5bAQsuTODB0U8KrERtqqxVkh1LvTLSDKkGsUrMXLMLzEnpQi2Nabg8Zn63[/tex]'], 'type': 102}
3
试明：若级数[tex=4.429x2.714]ySadpvq7BrEZCGdcnD6+aYYKMve96JE4sd3o4QbcQhyFEB8wySY9WvLLExFxYbd4[/tex]及[tex=4.0x2.714]ySadpvq7BrEZCGdcnD6+aZmrAzqw84m41cRzSztSWcBTzzE1dPl749vdarS2YDQV[/tex]皆收敛且[tex=11.071x1.286]KEk9vvqmh+CBNh5nnxa46HqdwXfcZ7FgwC5TRAiXnHCxUdO0AP0u57VybKAorlBU12GvShicsM4yb7ZMMzYXHg==[/tex]，则级数[tex=4.0x2.714]ySadpvq7BrEZCGdcnD6+aeoIzvM7G7HHNcjbk992lZRvUS0Xh/lotVqiUMKn/3kW[/tex]也收敛，若级数 [tex=1.571x1.357]hjG9g3d/QK7lH8zZx+2L/Q==[/tex]与[tex=1.5x1.357]QON3ZX8tDpa8hcC8KI55XA==[/tex]皆发散，问级数[tex=1.5x1.357]jW5HUp91iXA6ZnNyC/+CoQ==[/tex]的收敛性若何？提示：(1)先证级数[tex=5.5x2.714]ySadpvq7BrEZCGdcnD6+aRrMeIUlIOH1QB+K47pqlk3PJyHm4zPMgvDT8ndQ74qKuwSRORSNDNzjOlQ26lPWRw==[/tex]收敛。(2)可能收敛，也可能发散。例如[tex=11.643x2.357]ozp+yD1eymb4/EK2UAZZC9r19Jhg5MU9Rjw7GnxmzJSPaUrcZlSER0YjHa9TokIA11573J3i6NzMPGTxnbTueQ==[/tex][tex=5.643x1.357]NGWbQ9xjBbPjETUZQumFRw/fmak//iRLrHzvWXLffDA=[/tex]。
4
级数1 1/3 1/5 …收敛。