# Counterexamples around series (part 1)

Unless otherwise stated, $$(u_n)_{n \in \mathbb{N}}$$ and $$(v_n)_{n \in \mathbb{N}}$$ are two real sequences.

### If $$(u_n)$$ is non-increasing and converges to zero then $$\sum u_n$$ converges?

Is not true. A famous counterexample is the harmonic series $$\sum \frac{1}{n}$$ which doesn’t converge as $\displaystyle \sum_{k=p+1}^{2p} \frac{1}{k} \ge \sum_{k=p+1}^{2p} \frac{1}{2p} = 1/2,$ for all $$p \in \mathbb N$$.

### If $$u_n = o(1/n)$$ then $$\sum u_n$$ converges?

Does not hold as can be seen considering $$u_n=\frac{1}{n \ln n}$$ for $$n \ge 2$$. Indeed $$\int_2^x \frac{dt}{t \ln t} = \ln(\ln x) – \ln (\ln 2)$$ and therefore $$\int_2^\infty \frac{dt}{t \ln t}$$ diverges. We conclude that $$\sum \frac{1}{n \ln n}$$ diverges using the integral test. However $$n u_n = \frac{1}{\ln n}$$ converges to zero.

### If $$u_n = o(v_n)$$ and $$\sum v_n$$ converges then $$\sum u_n$$ converges?

Is not true as we can see taking $$v_n = \frac{(-1)^n}{n}$$ and $$u_n=\frac{1}{n \ln n}$$. $$\sum v_n$$ converges according to the alternating series test and $$\sum u_n$$ diverges according to previous paragraph. However $\lim\limits_{n \to \infty} \left\vert \frac{u_n}{v_n} \right\vert = \lim\limits_{n \to \infty} \frac{1}{\ln n} = 0.$

### If $$\sum u_n$$ is a positive converging series then $$\frac{u_{n+1}}{u_n}$$ is bounded?

Not either. Have a look at $u_n=\begin{cases} \frac{1}{n^2} & \text{for } n \text{ even} \\ \frac{1}{2^n} & \text{for } n \text{ odd.} \end{cases}$ Then for $$p \in \mathbb N$$ $\frac{u_{2p+1}}{u_{2p}} = \frac{2^{2p+1}}{4p^2}$ diverges to $$\infty$$. However $$\sum u_n$$ converges as $$\sum \frac{1}{n^2}$$ and $$\sum \frac{1}{2^n}$$ both converge.
{"ft_lang_label":"__label__en","ft_lang_prob":0.7166335,"math_prob":1.0000099,"size":1871,"snap":"2022-40-2023-06","text_gpt3_token_len":700,"char_repetition_ratio":0.16764863,"word_repetition_ratio":0.02255639,"special_character_ratio":0.39818278,"punctuation_ratio":0.06111111,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.00001,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-30T01:32:47Z\",\"WARC-Record-ID\":\"<urn:uuid:88a9b175-ce50-42f8-975e-2d20798d59fc>\",\"Content-Length\":\"65103\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d79abcac-0225-4db3-8103-47e556b9bcb4>\",\"WARC-Concurrent-To\":\"<urn:uuid:0a943bc1-35bf-47a4-a698-06701ff74b36>\",\"WARC-IP-Address\":\"107.180.248.46\",\"WARC-Target-URI\":\"https://www.mathcounterexamples.net/counterexamples-around-series-part-1/\",\"WARC-Payload-Digest\":\"sha1:4N53332RPFPNXA56ZA6AWTNWNKACJRQV\",\"WARC-Block-Digest\":\"sha1:F4UTQVTQABEELDD5IKTK5HKGHNBCQ5PC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499790.41_warc_CC-MAIN-20230130003215-20230130033215-00496.warc.gz\"}"}