How do you find the radius of convergence #Sigma x^n/3^n# from #n=[0,oo)#?

Answer 1

The series has radius on convergence #R=3#, that is:

#sum_(n=0)^oo x^n/3^n = 3/(3-x)# for #abs(x) < 3#

Writing the series as:

#sum_(n=0)^oo x^n/3^n = sum_(n=0)^oo (x/3)^n = sum_(n=0)^oo t^n#
where #t=x/3#.
Now this is the geometrical series of ratio #t# and we should know that it has radius of convergence #R=1#, so that it converges absolutely for:
#abs (t) < 1 => abs (x/3) < 1 => abs (x) <3#

and the total is:

#sum_(n=0)^oo x^n/3^n =1/(1-x/3)= 3/(3-x)#
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Answer 2

The radius of convergence is #=3#

The series ratio test is what we employ.

#∣a_(n+1)/a_n∣=(∣(x^(n+1)/(3^(n+1)))/(x^n/3^n)∣)#
#lim_(n->+oo)∣x^(n+1)/(x^n)*3^n/3^(n+1)∣#
#=∣x∣*1/3#

In order for the series to converge, we must

#∣x∣*1/3<1#

Consequently,

#∣x∣<3#
The series converge for # -3 < x <3#
For, #x=3#, #=>#, #sum_0^(+oo)3^n/3^n#
#=sum_0^(+oo)1#, the series diverge
For, #x=-3#, #=>#, #sum_0^(+oo)(-3)^n/3^n#
#=sum_0^(+oo)(-1)^n#, the series diverge
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Answer 3
To find the radius of convergence for the series \( \sum_{n=0}^{\infty} \frac{x^n}{3^n} \), you can use the ratio test. The ratio test states that if \( a_n \) is a sequence of non-zero numbers, then the series \( \sum_{n=0}^{\infty} a_n \) converges absolutely if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), and diverges if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1 \). If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \), the test is inconclusive. For the given series, \( a_n = \frac{x^n}{3^n} \). Applying the ratio test: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{x^{n+1}/3^{n+1}}{x^n/3^n} \right| = \lim_{n \to \infty} \left| \frac{x}{3} \right| \) For the series to converge, \( \left| \frac{x}{3} \right| < 1 \), which implies \( |x| < 3 \). Thus, the radius of convergence is 3.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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