设[tex=3.429x2.714]VLWXBs2tJukJHxqXwa0dqsztOiQpJW1T+cwMNiIYWks=[/tex]的收敛半径为[tex=7.286x1.357]BAgmDKaSxxtUjVDwzB+uOQhfzrmiCH+2us/KCp7QIoE=[/tex],并且在收敛圆周上一点绝对收敛。试证明这个级数对于所有的点 [tex=3.857x1.357]U1DbLD3J8TOxvSFX8XN6ncPbKLuFQZ6rnZ6hvDAIbnA=[/tex]为绝对收敛且一致收敛。
举一反三
- 设 [tex=3.714x3.286]WGu493lWbQkNjIXIJ06onT3QU0jn8OIdvbSEozq++L5iYU5MhSc0wTrCNvECtMFc[/tex]的收敛半径为[tex=2.643x1.071]32Pv2LkB2q4Pwe+441Rv4g==[/tex], 并且在收敛圆上一点绝对收敛. 试证明这个级数对于所有的点 [tex=3.857x1.357]CNBgVJ+m1syZQs4iWENU0nWdZQLGgCX0piG3F30ZMuM=[/tex]为绝对收敛.
- 设幂级数[tex=3.357x2.786]VCAPAvn3gOPyP36rvxBwz42vGlxEsX11yPxqpUctxqY=[/tex]的收敛区间为[tex=9.714x1.357]W/pJ7BbRMIyc+Ux1PFgVvQe9sUuGDFR0Jy3hIXtbaE0=[/tex],并且在[tex=3.5x1.143]6GgWoBVZ18MHpPu7z8SCmg==[/tex]处绝对收敛,证明它在[tex=3.357x1.357]aDH89bOKucg0qfwlTd+D8g==[/tex]上一致收敛。
- 设级数[tex=3.571x2.714]LCs/jzl+nr3KBTJXBn4IiTaMdvoS/p/hGL/Jv9ntegzmzbVBv3v1HeKEgBlLcyLM[/tex]的收敛半径为[tex=7.286x1.357]sTdvH6zX0iZNqILTrUec+Q==[/tex],证明:级数[tex=7.143x2.714]LCs/jzl+nr3KBTJXBn4IiVrR/a63aRDgwm6Ulx0DCkqQZXUGezi8qQqRicSofTkUWyb3f6mqFTz2twehW0bB7Q==[/tex]的收敛半径为[tex=7.714x2.786]88n1NtKriG0YM72QT5w50ARKMC3GTCC7OGxgHAYlBdhewQuMEfErAwKQ9wpi7IxVHJewFuEn04JodLhBCFDNBA==[/tex].
- 求证:级数[tex=4.143x3.286]3PXegz5bAQsuTODB0U8KrO8dE2QFyGzTKIgAWyUOAjW2NnK99u1z9bgI+kTvhzvW[/tex]在[tex=2.857x1.357]bxkuEf5OdjtfMlwpIXZhFYFm4mzHICAh/+PdGm82/Ds=[/tex]上发散;在[tex=2.857x1.357]W2UvKR01GUJgbq0KdXYJYQ==[/tex]内绝对收敛且内闭 一致收敛,但非一致收敛。
- 设幂级数[tex=3.643x3.286]WGu493lWbQkNjIXIJ06onV6IZmMDrYShNGcPME8shwWKH5T1GYVkFqbYkQtvxgXS[/tex]的收敛半径为[tex=1.143x1.214]WB5oUFU97imVoOqmwwnMtg==[/tex], 而[tex=3.5x3.429]UU1qstNjdmzg7TFKGbeGXsJXpXGu4k7SZ5Pl374mxwk=[/tex]的收敛半径为[tex=1.143x1.214]akFdfHl3PdcRxRUQleHWdA==[/tex].若把幂级数[tex=6.214x3.286]WGu493lWbQkNjIXIJ06oneLZcJFoQ3BGITMlybWara2JPRKknBTl8nFXbTZweoPu0vBt34L3pxIcH/n/A76GVQ==[/tex]的收敛半径记为[tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex], 证明:[tex=7.286x1.357]/Ormn0xncvBSYPuYSYE8Zf4KYeLykBmiGoKt1A6m2PKY9SnlqBOnZ0Or2B4jHlMy[/tex]