Consider the series:
"\u221e"
"\u03a3" [ "(x^2 + 9)" / 25 "]^a"
"a = 1"
The values of "y" of which this series converges form an internal "(-k, k)" .
Give the value for "k"
It is given that,
"\\displaystyle\\sum_{a=1}^\\infin\\lbrack\\frac{(x^2+9)}{25}\\rbrack^a=\\displaystyle\\sum_{a=1}^\\infin\\lbrack M_a\\rbrack"
"\\implies M_a=\\lbrack\\frac{(x^2+9)}{25}\\rbrack^a"
By the ratio test,
"L=\\lim_{a\\rightarrow \\infty} \\vert\\frac{M_{a+1}}{M_a}\\vert"
"L=\\lim_{a\\rightarrow \\infty }\\vert\\frac{\\lbrack\\frac{ (x^2+9)}{25}\\rbrack ^{a+1}}{\\lbrack \\frac{(x^2+9)}{25}\\rbrack^a}\\vert"
"L=lim_{ a\\rightarrow \\infty}\\space \\vert\\lbrack\\frac{(x^2+9)}{25}\\rbrack\\vert"
"L=\\vert \\lbrack\\frac{(x^2+9)}{25}\\rbrack\\vert"
For interval of convergence,
L<1
"\\vert\\frac{(x^2+9)}{25}\\vert<1"
"\\frac{x^2+9}{25}<1"
"x^2+9<25\\\\x^2+9-25<0\\\\x^2-16<0\\\\(x-4)(x+4)<0\\\\-4<x<4"
Hence the interval of convergence is "x\\in(-4,4)"
and "k=4"
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