数列 lim n趋于+∞ 1+1/2+1/4...+1/2^n / 1+1/3+1/9...+1/3^n 这个数列的极限是多少,还有个 lim n→+∞ (2n+1)^4-(n-1)^n / (n+5)^4+(3n+1)^4
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![数列 lim n趋于+∞ 1+1/2+1/4...+1/2^n / 1+1/3+1/9...+1/3^n 这个数列的极限是多少,还有个 lim n→+∞ (2n+1)^4-(n-1)^n / (n+5)^4+(3n+1)^4](/uploads/image/z/15253709-5-9.jpg?t=%E6%95%B0%E5%88%97+lim+n%E8%B6%8B%E4%BA%8E%2B%E2%88%9E+1%2B1%2F2%2B1%2F4...%2B1%2F2%5En+%2F+1%2B1%2F3%2B1%2F9...%2B1%2F3%5En+%E8%BF%99%E4%B8%AA%E6%95%B0%E5%88%97%E7%9A%84%E6%9E%81%E9%99%90%E6%98%AF%E5%A4%9A%E5%B0%91%2C%E8%BF%98%E6%9C%89%E4%B8%AA+lim+n%E2%86%92%2B%E2%88%9E+%EF%BC%882n%2B1%29%5E4-%28n-1%29%5En+%2F+%28n%2B5%29%5E4%2B%283n%2B1%29%5E4)
数列 lim n趋于+∞ 1+1/2+1/4...+1/2^n / 1+1/3+1/9...+1/3^n 这个数列的极限是多少,还有个 lim n→+∞ (2n+1)^4-(n-1)^n / (n+5)^4+(3n+1)^4
数列 lim n趋于+∞ 1+1/2+1/4...+1/2^n / 1+1/3+1/9...+1/3^n 这个数列的极限是多少,
还有个 lim n→+∞ (2n+1)^4-(n-1)^n / (n+5)^4+(3n+1)^4
数列 lim n趋于+∞ 1+1/2+1/4...+1/2^n / 1+1/3+1/9...+1/3^n 这个数列的极限是多少,还有个 lim n→+∞ (2n+1)^4-(n-1)^n / (n+5)^4+(3n+1)^4
1.求极限 n→+∞lim[(1+1/2+1/4...+1/2ⁿ) / (1+1/3+1/9...+1/3ⁿ)]
右边分子是首项为1,公比为1/2的等比数列;分母是首项为1,公比为1/3的等比数列;故:
原式=n→+∞lim[2(1-2ⁿ)/(3/2)(1-1/3ⁿ)]=2/(3/2)=4/3.
2.求极限 n→+∞lim[ (2n+1)⁴-(n-1)ⁿ] / [(n+5)⁴+(3n+1)⁴]
原式= n→+∞lim{(2n+1)⁴/[(n+5)⁴+(3n+1)⁴]-(n-1)ⁿ/[(n+5)⁴+(3n+1)⁴]}
=n→+∞lim(2n)⁴/[n⁴+(3n)⁴]- {n→+∞lim(n-1)ⁿ/[(n+5)⁴+(3n+1)⁴]}
=16/82-{n→+∞lim[nⁿ/(n⁴+81n⁴)}=8/41-{n→+∞lim[nⁿֿ⁴/82]}=-∞
如果原题分母上的两项有一项的指数不是4,而是n,那么结果就不一样了!
n→+∞lim[ (2n+1)⁴-(n-1)ⁿ] / [(n+5)⁴+(3n+1)ⁿ]
=n→+∞lim(16n⁴-nⁿ)/[n⁴+(3n)ⁿ]=n→+∞lim(16-nⁿֿ⁴)/(1+3ⁿnⁿֿ⁴)
=n→+∞lim[(16/nⁿֿ⁴)-1]/[(1/nⁿֿ⁴)+3ⁿ]=0
lim n趋于+∞ 1+1/2+1/4...+1/2^n / 1+1/3+1/9...+1/3^n
=limn趋于+∞ 1+1/2+1/4...+1/2^n+ lim n趋于+∞1+1/3+1/9...+1/3^n
=lim1/(1-1/2)+lim1/(1-1/3)
=2+3/2
=7/2