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o-na [289]
3 years ago
7

Use root test to determine the series is convergent/divergent or inconclusive

Mathematics
1 answer:
Alinara [238K]3 years ago
8 0

You can first condense the series to make it simpler to study. If <em>n</em> is odd, then <em>n</em> = 2<em>k</em> - 1, and if <em>n</em> is even, then <em>n</em> = 2<em>k</em> for some <em>k</em> ≥ 1. So for each <em>k</em>, you can pair up the <em>k</em>-th terms of the odd- and even-indexed series.

odd:

\dfrac1{3^{\frac{n+3}2}}=\dfrac1{3^{k+1}}

even:

\dfrac1{3^{\frac n2}}=\dfrac1{3^k}

So the series can be re-indexed as

\displaystyle\sum_{n=1}^\infty a_n=\sum_{k=1}^\infty a_k=\sum_{k=1}^\infty \left(\frac1{3^{k+1}}+\frac1{3^k}\right)=\sum_{k=1}^\infty\frac4{3^{k+1}}

By the root test, the series converges, since

\displaystyle\lim_{k\to\infty}\sqrt[k]{\left|\frac4{3^{k+1}}\right|}=\frac13\lim_{k\to\infty}\left(\frac43\right)^{\frac1k}=\frac13

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In the equation a= 3/b is a positive, real number. As the
Igoryamba

Answer:

<h3>The value of a tends to zero</h3>

Step-by-step explanation:

Given the expression a = 3/b

as the denominator of the of the expression b increases the value of the output is known to be decreasing. if the value of b now goes so large to become an infinite, value then the value of a will goes to zero as shown;

If b = ∞

a = 3/∞

a = 0

Note that for any constant value a, the ratio a/∞ will always tends to zero

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3 years ago
in a favorite punch mix there are 3 parts orange juice, 2 parts pineapple juice and 1 part ginger ale. How many quarts of each w
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Number of parts of the drinks = 31+1 = 6

In a 30 quart mixture:-

amount of orange juice = 3/6 + 30  =  15 qrts
   ....  ... ...  pineapple    = 2/6 * 30   =  10 qrts
  ....  ......... ginger ale    = 1/6 * 30  =    5 qrts
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2 years ago
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pashok25 [27]
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21.(1/2)/(1/72)=36
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3 years ago
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Karo-lina-s [1.5K]

\text{The scientific notation:}\\\\a\times10^n\\\\1\leq a


a.\\(5\times10^3)\times(3.5\times10^{-1})=(5\times3.5)\times(10^3\times10^{-1})\\\\=15.75\times10^{3+(-1)}=15.75\times10^2=1.575\times10\times10^2\\\\=1.575\times10^{1+2}=\boxed{1.575\times10^3}

b.\\(8\times10^2)\div(4\times10^3)=(8\div4)\times(10^2\div10^3)=2\times10^{2-3}\\\\=\boxed{2\times10^{-1}}

c.\\(6\times10^{-3})^2=6^2\times(10^{-3})^2=36\times10^{-3\cdot2}=36\times10^{-6}=3.6\times10\times10^{-6}\\\\=3.6\times10^{1+(-6)}=\boxed{3.6\times10^{-5}}

Used:\\\\a^n\times a^m=a^{n+m}\\\\a^n\div a^m=a^{n-m}\\\\(a^n)^m=a^{n\cdot m}

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3 years ago
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