true or false,
1) If lim n→∞, an = 0, then Σan is convergent.
2) If lim n→∞, an = 5, then lim n→∞ a n+5 = 5 is convergent.
1) No. Consider the harmonic series
2) ??? if the nth term has a limit, then it converges to that limit.
so a_n+5 also converges to that limit.
So, the first one is false and second one is true?
1) The statement is false. The sum of the terms of a series, denoted by Σan, can converge or diverge regardless of whether the limit of the individual terms an approaches zero. To determine whether the series converges, we need to examine the behavior of the partial sums.
To illustrate this, consider the harmonic series: Σ(1/n). Although the terms approach zero as n→∞, the series diverges because the partial sums keep growing without bound.
2) The statement is true. If the limit of the sequence an is a constant value, such as 5 in this case, then the limit of the shifted sequence an+5 would also be 5.
To understand why this is the case, consider that when n approaches infinity, adding a constant value to n does not affect the limit. So, as n→∞, an+5 is essentially evaluating the limit of the original sequence an but with n shifted by 5. Hence, the limit of an+5 would also be 5, making it convergent.
In general, when dealing with limits, shifting the index of the sequence by a constant does not change the limit.