14. Sequences & Series

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(−1)ⁿ / 2ⁿ}

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \$100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \$100.

Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.

a.Write the first five terms of the sequence {Bₙ}.

The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.

n an

1 2.59374246

2 2.70481383

3 2.71692393

4 2.71814593

5 2.71826824

6 2.71828047

7 2.71828169

8 2.71828179

9 2.71828204

10 2.71828203